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(subsection.2.1) cvn /Title (The tree T) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (subsection.2.2) cvn /Title (Shortest walks in T) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (subsection.2.3) cvn /Title (The tree R) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (subsection.2.4) cvn /Title (Projection of walks of T into walks of R) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -4 /Dest (section.3) cvn /Title (Proof of Theorem 1) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (subsection.3.1) cvn /Title (Step 1: the growth rate of n\(a,b\) does not depend on \(a,b\)) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (subsection.3.2) cvn /Title (Step 2: an explicit expression of n\(a,b\) in terms of k\(|b-a|,a\) for \(a,b\)=\(1,\) or \(1,-1\)) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (subsection.3.3) cvn /Title (Step 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Ft(suc)-36 b(h)450 b(that)f Fr(r)43151 46847 y Fy(i)43925 46648 y Ft(=)397 b Fr(n)46110 46847 y Fy(i)46486 46648 y Fr(r)47072 46847 y Fy(i)p Fv(+1)48956 46648 y Ft(+)307 b Fr(r)50861 46847 y Fy(i)p Fv(+2)52439 46648 y Ft(.)800 48253 y(Let)414 b(us)h(denote)f(b)-36 b(y)414 b Fr(N)553 b Ft(the)414 b(index)h(suc)-36 b(h)414 b(that)g Fr(r)24323 48452 y Fy(N)25589 48253 y Ft(=)368 b(0.)573 b(Since)414 b Fr(a)g Ft(and)h Fr(b)f Ft(are)h(relativ)-36 b(ely)416 b(prime,)j(w)-36 b(e)415 b(ha)-36 b(v)g(e)800 49858 y Fr(r)1386 50057 y Fy(N)94 b Fw(\241)p Fv(1)3854 49858 y Ft(=)369 b(1)402 b(so,)409 b(thanks)402 b(to)g(Lemma)p 0 .5 0 TeXcolorrgb 18592 49858 a SDict begin H.S end 18592 49858 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 19242 49002 a SDict begin H.R end 19242 49002 a 19242 49858 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.1) cvn H.B /ANN pdfmark end 19242 49858 a Black Ft(,)409 b(the)402 b(pairs)g(\()p Fr(r)26468 50057 y Fy(N)94 b Fw(\241)p 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end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Ft(the)485 b(pair)h(\()p Fr(r)6910 -2473 y Fy(i)7285 -2672 y Fr(;)221 b(r)8453 -2473 y Fy(i)p Fw(\241)p Fv(1)10032 -2672 y Ft(\).)734 b(If)486 b Fr(n)13761 -2473 y Fy(i)p Fw(\241)p Fv(1)15825 -2672 y Ft(is)g(o)36 b(dd,)498 b(w)-36 b(e)486 b(get)g(the)f(pair)g(\()p Fr(r)30472 -2473 y Fy(i)30848 -2672 y Fr(;)221 b(r)32016 -2473 y Fy(i)p Fw(\241)p Fv(1)33925 -2672 y Fq(\241)331 b Fr(r)35875 -2473 y Fy(i)36251 -2672 y Ft(\).)734 b(An)485 b(addition)g(follo)-36 b(w)g(ed)487 b(b)-36 b(y)486 b(a)800 -1067 y(subtraction)433 b(then)g(giv)-36 b(es)434 b(the)f(pair)h(\()p Fr(r)19827 -868 y Fy(i)p Fw(\241)p Fv(1)21405 -1067 y Fr(;)221 b(r)22573 -868 y Fy(i)22949 -1067 y Ft(\).)2751 539 y(In)473 b(conclusion,)484 b(starting)474 b(from)g(\()p Fr(r)20213 738 y Fy(i)p Fv(+1)21791 539 y Fr(;)221 b(r)22959 738 y Fy(i)23336 539 y Ft(\))473 b(and)g(\()p Fr(r)27976 738 y Fy(i)28352 539 y Fr(;)221 b(r)29520 738 y Fy(i)p Fv(+1)31098 539 y Ft(\))474 b(resp)36 b(ectiv)-36 b(ely)-108 b(,)484 b(w)-36 b(e)474 b(obtain)g(\()p Fr(r)46603 738 y Fy(i)46978 539 y Fr(;)221 b(r)48146 738 y Fy(i)p Fw(\241)p Fv(1)49725 539 y Ft(\))474 b(and)800 2144 y(\()p Fr(r)1892 2343 y Fy(i)p Fw(\241)p Fv(1)3470 2144 y Fr(;)221 b(r)4638 2343 y Fy(i)5015 2144 y Ft(\))545 b(if)g Fr(n)8145 2343 y Fy(i)p Fw(\241)p Fv(1)10269 2144 y Ft(is)h(o)36 b(dd,)573 b(\()p Fr(r)15847 2343 y Fy(i)p Fw(\241)p Fv(1)17425 2144 y Fr(;)221 b(r)18593 2343 y Fy(i)18969 2144 y Ft(\))545 b(and)g(\()p Fr(r)23753 2343 y Fy(i)24129 2144 y Fr(;)221 b(r)25297 2343 y Fy(i)p Fw(\241)p Fv(1)26875 2144 y Ft(\))545 b(if)h Fr(n)30006 2343 y Fy(i)p Fw(\241)p Fv(1)32130 2144 y Ft(is)f(ev)-36 b(en,)574 b(and)545 b(the)f(prop)36 b(osition)546 b(is)f(th)-36 b(us)800 3749 y(pro)g(v)g(ed)433 b(b)-36 b(y)434 b(induction.)p 51860 3749 45 878 v 51905 2915 781 45 v 51905 3749 V 52684 3749 45 878 v 2751 6350 a(Recall)k(that)e Fq(b)p Fr(x)p Fq(c)h Ft(denotes)f(the)g(in)-36 b(teger)437 b(part)f(of)h Fr(x)p Ft(.)587 b(W)-108 b(e)437 b(can)f(sum)g(up)g(the)g(previous)h(construction)800 7955 y(in)c(the)h(follo)-36 b(wing)435 b(w)-36 b(a)g(y:)p Black 2737 10999 a Fq(\262)p Black 651 w Ft(if)434 b Fr(n)6020 11198 y Fy(i)p Fw(\241)p Fv(1)8032 10999 y Ft(is)g(o)36 b(dd,)433 b(then)p Black 5516 14043 a Fo({)p Black 650 w Ft(starting)h(from)g(\()p Fr(r)15925 14242 y Fy(i)16300 14043 y Fr(;)221 b(r)17468 14242 y Fy(i)p Fv(+1)19047 14043 y Ft(\),)434 b(w)-36 b(e)434 b(attain)g(\()p Fr(r)27185 14242 y Fy(i)p Fw(\241)p Fv(1)28762 14043 y Fr(;)221 b(r)29930 14242 y Fy(i)30307 14043 y Ft(\))433 b(in)h(2)295 b(+)g(3)h Fq(\242)f(b)p Fr(n)37992 14242 y Fy(i)p Fw(\241)p Fv(1)39571 14043 y Fr(=)p Ft(2)p Fq(c)434 b Ft(steps;)p Black 5516 16202 a Fo({)p Black 650 w Ft(starting)g(from) g(\()p Fr(r)15925 16401 y Fy(i)p Fv(+1)17503 16202 y Fr(;)221 b(r)18671 16401 y Fy(i)19047 16202 y Ft(\),)434 b(w)-36 b(e)434 b(attain)g(\()p Fr(r)27185 16401 y Fy(i)27560 16202 y Fr(;)221 b(r)28728 16401 y Fy(i)p Fw(\241)p Fv(1)30307 16202 y Ft(\))433 b(in)h(1)295 b(+)g(3)h Fq(\242)f(b)p Fr(n)37992 16401 y Fy(i)p Fw(\241)p Fv(1)39571 16202 y Fr(=)p Ft(2)p Fq(c)434 b Ft(steps;)p Black 2737 19246 a Fq(\262)p Black 651 w Ft(if)g Fr(n)6020 19445 y Fy(i)p Fw(\241)p Fv(1)8032 19246 y Ft(is)g(ev)-36 b(en,)434 b(then)p Black 5516 22290 a Fo({)p Black 650 w Ft(starting)g(from)g(\() p Fr(r)15925 22489 y Fy(i)16300 22290 y Fr(;)221 b(r)17468 22489 y Fy(i)p Fv(+1)19047 22290 y Ft(\),)434 b(w)-36 b(e)434 b(attain)g(\()p Fr(r)27185 22489 y Fy(i)27560 22290 y Fr(;)221 b(r)28728 22489 y Fy(i)p Fw(\241)p Fv(1)30307 22290 y Ft(\))433 b(in)h(3)p Fr(n)34190 22489 y Fy(i)p Fw(\241)p Fv(1)35769 22290 y Fr(=)p Ft(2)g(steps;)p Black 5516 24449 a Fo({)p Black 650 w Ft(starting)g(from)g(\()p Fr(r)15925 24648 y Fy(i)p Fv(+1)17503 24449 y Fr(;)221 b(r)18671 24648 y Fy(i)19047 24449 y Ft(\),)434 b(w)-36 b(e)434 b(attain)g(\()p Fr(r)27185 24648 y Fy(i)p Fw(\241)p Fv(1)28762 24449 y Fr(;)221 b(r)29930 24648 y Fy(i)30307 24449 y Ft(\))433 b(in)h(3)p Fr(n)34190 24648 y Fy(i)p Fw(\241)p Fv(1)35769 24449 y Fr(=)p Ft(2)g(steps.)2751 27493 y(It)477 b(is)g(easily)i(seen)d(that)h(if)h(the)e(pair)h(\()p Fr(a;)221 b(b)p Ft(\))477 b(app)36 b(ears)477 b(in)g Fo(T)31911 27700 y Fv(\()p Fy(c;d)p Fv(\))33851 27493 y Ft(,)488 b(then)476 b(the)h(full)g(tree)g Fo(T)45993 27700 y Fv(\()p Fy(a;b)p Fv(\))48421 27493 y Ft(app)36 b(ears)800 29098 y(in)498 b Fo(T)3421 29305 y Fv(\()p Fy(c;d)p Fv(\))5362 29098 y Ft(,)515 b(in)498 b(an)h(ob)-36 b(vious)499 b(sense.)773 b(The)498 b(follo)-36 b(wing)501 b(sho)-36 b(ws)498 b(a)h(\\self-con)-36 b(tainmen)g(t")499 b(asp)36 b(ect)498 b(of)h(random)800 30703 y(Fib)36 b(onacci)434 b(trees.)800 30703 y SDict begin H.S end 800 30703 a 800 30703 a SDict begin 14.5 H.A end 800 30703 a 800 30703 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.2) cvn H.B /DEST pdfmark end 800 30703 a Black 2712 x Fo(Prop)42 b(osition)531 b(2.)p Black 567 w Fs(F)-100 b(or)491 b(any)e(p)-66 b(ositive)489 b(inte)-66 b(gers)487 b Fr(a)j Fs(and)g Fr(b)f Fs(r)-66 b(elatively)488 b(prime,)495 b Fo(T)40687 33622 y Fv(\()p Fy(a;b)p Fv(\))43128 33415 y Fs(app)-66 b(e)g(ars)490 b(in\257nitely)800 35020 y(many)464 b(times)g(in)g Fo(T)p Fs(.)p Black 800 37732 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 11784 37732 a SDict begin H.S end 11784 37732 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 12435 36876 a SDict begin H.R end 12435 36876 a 12435 37732 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.1) cvn H.B /ANN pdfmark end 12435 37732 a Black 347 w Ft(sho)-36 b(ws)348 b(that)f(an)-36 b(y)348 b(pair)g(of)g(the)f(form)h(\()p Fr(a;)221 b(b)p Ft(\))348 b(where)f Fr(a)h Ft(and)f Fr(b)g Ft(are)h(relativ)-36 b(ely)349 b(prime)800 39337 y(app)36 b(ears)502 b(in)h Fo(T)p Ft(.)785 b(It)502 b(is)h(then)e(enough)h(to)h(sho)-36 b(w)502 b(that)g Fo(T)h Ft(app)36 b(ears)502 b(in)g Fo(T)36772 39544 y Fv(\()p Fy(a;b)p Fv(\))38723 39337 y Ft(.)785 b(W)-108 b(e)503 b(consider)f(a)g(random)800 40943 y(Fib)36 b(onacci)471 b(sequence)g(\()p Fr(g)13337 41142 y Fy(n)13963 40943 y Ft(\))14469 41142 y Fy(n)15566 40943 y Ft(suc)-36 b(h)470 b(that)g Fr(g)22015 41142 y Fv(0)22973 40943 y Ft(=)431 b Fr(g)25039 41142 y Fv(1)25997 40943 y Ft(=)h(1,)481 b Fr(g)29556 41142 y Fy(n)p Fw(\241)p Fv(1)31816 40943 y Ft(=)432 b Fr(a)471 b Ft(and)f Fr(g)37603 41142 y Fy(n)38661 40943 y Ft(=)432 b Fr(b)471 b Ft(for)g(an)g Fr(n)p Ft(.)690 b(F)-108 b(or)470 b Fr(k)478 b(>)431 b Ft(0,)800 42548 y(w)-36 b(e)434 b(then)f(de\257ne)f Fr(g)10058 42747 y Fy(n)p Fv(+)p Fy(k)12363 42548 y Ft(as)i Fq(j)p Fr(g)14952 42747 y Fy(n)p Fv(+)p Fy(k)24 b Fw(\241)p Fv(1)18321 42548 y Fq(\241)295 b Fr(g)20272 42747 y Fy(n)p Fv(+)p Fy(k)24 b Fw(\241)p Fv(2)23346 42548 y Fq(j)p Ft(.)2751 44153 y(It)332 b(is)g(easily)i(seen)e(that,)352 b(for)332 b(an)-36 b(y)332 b(in)-36 b(tegers)332 b Fr(u)h Ft(and)e Fr(v)48 b Ft(,)352 b(w)-36 b(e)332 b(alw)-36 b(a)g(ys)334 b(ha)-36 b(v)g(e)332 b(max)q(\()p Fq(j)p Fr(v)135 b Fq(\241)88 b Fr(u)p Fq(j)p Fr(;)221 b(v)48 b Ft(\))369 b Fq(\267)g Ft(max)q(\()p Fr(v)48 b(;)221 b(u)p Ft(\),)800 45758 y(with)325 b(equalit)-36 b(y)326 b(i\256)f(these)g(maxima)h(are)f(b)36 b(oth)325 b(equal)h(to)f Fr(v)48 b Ft(.)541 b(The)326 b(equalit)-36 b(y)326 b(case)f(cannot,)347 b(therefore,)g(o)36 b(ccur)800 47363 y(for)454 b(t)-36 b(w)g(o)453 b(successiv)-36 b(e)453 b(pairs)g(of)h Fr(g)16651 47562 y Fy(n)p Fv(+)p Fy(k)18523 47363 y Ft(.)636 b(As)453 b(a)g(consequence,)459 b(w)-36 b(e)453 b(get)g(that)g(there)f(is)h(a)g Fr(k)447 b Fq(\267)403 b Ft(2)p Fr(n)453 b Ft(suc)-36 b(h)452 b(that)800 48968 y Fr(g)1423 49167 y Fy(n)p Fv(+)p Fy(k)3663 48968 y Ft(=)369 b Fr(g)5667 49167 y Fy(n)p Fv(+)p Fy(k)24 b Fv(+1)9110 48968 y Ft(=)368 b(1,)435 b(and)e(w)-36 b(e)433 b(are)h(done.)p 51860 48968 45 878 v 51905 48134 781 45 v 51905 48968 V 52684 48968 45 878 v 2751 51569 a(As)g(a)g(corollary)-108 b(,)435 b(w)-36 b(e)434 b(get)f(the)g(follo) -36 b(wing)436 b(result:)800 51828 y SDict begin H.S end 800 51828 a 800 51828 a SDict begin 14.5 H.A end 800 51828 a 800 51828 a SDict begin [ /View [/XYZ H.V] /Dest (Cor.1) cvn H.B /DEST pdfmark end 800 51828 a Black 2453 x Fo(Corollary)521 b(1.)p Black 554 w Fs(A)-33 b(ny)463 b(p)-66 b(air)465 b(app)-66 b(e)g(aring)463 b(in)h Fo(T)h Fs(app)-66 b(e)g(ars)465 b(in\257nitely)e(many)h(times)g(in)g Fo(T)p Fs(.)2751 56993 y Ft(Therefore,)388 b(if)377 b Fr(a)e Ft(and)h Fr(b)g Ft(are)g(relativ)-36 b(ely)377 b(prime,)388 b(the)375 b(random)g(Fib)36 b(onacci)377 b(tree)e(generated)h(b)-36 b(y)375 b Fr(a)h Ft(with)800 58599 y(a)530 b(c)-36 b(hild)530 b Fr(b)g Ft(is)h(a)f(subtree)f(of)i Fo(T)p Ft(.)868 b(Recall)531 b(that,)554 b(if)531 b(this)f(is)g(not)g 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1.625 1.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.875 1.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 2.125 2.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 2.375 2.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1574 w @beginspecial @setspecial 2.625 2.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 2.875 2.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 3.125 3.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 3.375 3.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 3.625 3.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 3.875 3.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 4.125 4.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.375 4.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 4.625 4.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.875 4.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1574 w @beginspecial @setspecial 5.125 5.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.375 5.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 5.625 5.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.875 5.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 6.125 6.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 6.375 6.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 6.625 6.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 6.875 6.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 7.125 7.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 7.375 7.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1574 w @beginspecial @setspecial 7.625 7.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 7.875 7.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 8.125 8.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 8.375 8.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 8.625 8.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 8.875 8.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 35886 6299 a @beginspecial @setspecial 12 13 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14 13 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 12598 w @beginspecial @setspecial 16 17 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 18 17 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 32736 9449 a @beginspecial @setspecial 11.5 12 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.5 12 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6300 w @beginspecial @setspecial 13.5 14 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.5 14 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6299 w @beginspecial @setspecial 15.5 16 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.5 16 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6299 w @beginspecial @setspecial 17.5 18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 18.5 18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 31161 12598 a @beginspecial @setspecial 11.25 11.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 11.75 11.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 w @beginspecial @setspecial 12.25 12.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.75 12.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 w @beginspecial @setspecial 13.25 13.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.75 13.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3149 w @beginspecial @setspecial 14.25 14.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.75 14.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 w @beginspecial @setspecial 15.25 15.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.75 15.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 w @beginspecial @setspecial 16.25 16.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.75 16.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3149 w @beginspecial @setspecial 17.25 17.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 17.75 17.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 w @beginspecial @setspecial 18.25 18.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 18.75 18.5 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 30374 15748 a @beginspecial @setspecial 11.125 11.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 11.375 11.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 11.625 11.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 11.875 11.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 12.125 12.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.375 12.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1574 w @beginspecial @setspecial 12.625 12.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.875 12.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 13.125 13.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.375 13.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 13.625 13.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.875 13.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 14.125 14.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.375 14.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 14.625 14.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.875 14.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 15.125 15.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.375 15.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1574 w @beginspecial @setspecial 15.625 15.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.875 15.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 16.125 16.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.375 16.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 16.625 16.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.875 16.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 17.125 17.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 17.375 17.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 17.625 17.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 17.875 17.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1574 w @beginspecial @setspecial 18.125 18.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 18.375 18.25 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1575 w @beginspecial @setspecial 18.625 18.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 18.875 18.75 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 26437 -3150 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 x @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 10689 3150 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 31496 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4390 6299 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 12598 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1240 9449 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6299 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6300 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6299 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -335 12598 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3149 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3149 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -1122 15748 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1574 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1574 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1574 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -1516 18898 a @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 35886 6299 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 12598 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 32736 9449 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6300 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6299 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6299 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 31161 12598 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3149 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3149 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 30374 15748 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1574 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1574 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1574 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1575 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 29980 18898 a @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 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@beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 787 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 788 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 26112 -2722 a Ft(1)26112 428 y(1)10364 3578 y(0)30846 b(2)4065 6727 y(1)11948 b(1)915 9877 y(1)5649 b(1)g(1)h(1)-660 13026 y(0)2500 b(2)f(0)h(2)g(0)f(2)h(0)g(2)-1447 16176 y(1)925 b(1)f(1)h(3)g(1)g(1)g(1)f(3)h(1)g(1)g(1)g(3)g(1)f(1)h(1)g(3) -1719 19112 y Fl(1)382 b(1)f(1)h(1)f(1)g(3)h(1)f(5)h(1)f(1)g(1)h(1)f(1) h(3)f(1)g(5)h(1)f(1)h(1)f(1)g(1)h(3)f(1)h(5)f(1)g(1)h(1)f(1)h(1)f(3)g (1)h(5)35561 6727 y Ft(1)11948 b(3)32411 9877 y(1)5649 b(3)h(1)f(5)30836 13026 y(0)2500 b(2)g(2)f(4)h(2)f(4)h(2)g(8)30049 16176 y(1)925 b(1)g(1)f(3)h(1)g(5)g(1)g(7)f(1)h(3)g(3)g(5)g(3)f(7)h(3) 600 b(13)29777 19112 y Fl(1)382 b(1)f(1)h(1)f(1)g(3)h(1)f(5)h(1)f(3)g (3)h(7)f(3)h(5)f(3)179 b(11)f(1)381 b(3)h(1)f(5)g(1)h(7)f(1)h(9)f(1)g (5)h(5)f(9)h(5)178 b(11)g(5)h(21)800 22047 y SDict begin H.S end 800 22047 a 800 22047 a SDict begin 14.5 H.A end 800 22047 a 800 22047 a SDict begin [ /View [/XYZ H.V] /Dest (figure.1) cvn H.B /DEST pdfmark end 800 22047 a Black 12619 26364 a Ft(Figure)433 b(1:)p 0 TeXcolorgray 18309 24759 a SDict begin H.S end 18309 24759 a 0 TeXcolorgray 0 TeXcolorgray 18309 24759 a SDict begin H.R end 18309 24759 a 18309 24759 a SDict begin [ /View [/XYZ H.V] /Dest (figure.1) cvn H.B /DEST pdfmark end 18309 24759 a Black 18743 26364 a Ft(The)g(random)g(Fib)36 b(onacci)435 b(tree)e Fo(T)369 b Ft(=)g Fo(T)38558 26571 y Fv(\(1)p Fy(;)p Fv(1\))p Black Black 2751 29947 a Ft(Let)423 b(us)g(men)-36 b(tion,)425 b(without)e(further)g(elab)36 b(oration,)427 b(that)422 b(the)h(sequence)h(of)g(lab)36 b(els)424 b(in)f(the)g(tree)g (read)800 31552 y(in)488 b(breadth-\257rst)e(order)i(\(1,)502 b(1,)h(0,)f(2,)h(1,)f(1,)h(1,)f(3,)h(1,)g(1,)f(1,)h(1,)f(1,)h(3,)f(1,)h (5.)217 b(.)g(.)g(\),)502 b(giv)-36 b(es)490 b(an)e(example)h(of)g(a) 800 33157 y Fs(2-r)-66 b(e)g(gular)516 b(se)-66 b(quenc)g(e)488 b Ft(in)i(the)f(terminology)i(giv)-36 b(en)490 b(b)-36 b(y)490 b(Allouc)-36 b(he)490 b(and)f(Shallit)h([)p 0 .5 0 TeXcolorrgb 40558 33157 a SDict begin H.S end 40558 33157 a 0 .5 0 TeXcolorrgb Ft(2)p 0 .5 0 TeXcolorrgb 41209 32301 a SDict begin H.R end 41209 32301 a 41209 33157 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.AS1) cvn H.B /ANN pdfmark end 41209 33157 a Black Ft(,)p 0 .5 0 TeXcolorrgb 42060 33157 a SDict begin H.S end 42060 33157 a 0 .5 0 TeXcolorrgb Ft(3)p 0 .5 0 TeXcolorrgb 42710 32301 a SDict begin H.R end 42710 32301 a 42710 33157 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.AS2) cvn H.B /ANN pdfmark end 42710 33157 a Black Ft(])g(\(see)g(also)h(section)p 0 .5 0 TeXcolorrgb 800 34763 a SDict begin H.S end 800 34763 a 0 .5 0 TeXcolorrgb Ft(2.3)p 0 .5 0 TeXcolorrgb 2462 33906 a SDict begin H.R end 2462 33906 a 2462 34763 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.2.3) cvn H.B /ANN pdfmark end 2462 34763 a Black 434 w Ft(of)434 b(the)f(presen)-36 b(t)432 b(article,)j(after)f(Figure)f(2\).)800 36953 y SDict begin H.S end 800 36953 a 800 36953 a SDict begin 14.5 H.A end 800 36953 a 800 36953 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.2) cvn H.B /DEST pdfmark end 800 36953 a 1661 x Fn(2.2)1793 b(Shortest)599 b(w)-50 b(alks)599 b(in)f(T)800 41077 y Ft(By)490 b(a)f(shortest)f(w)-36 b(alk)491 b(from)e(the)f(ro)36 b(ot)490 b(of)f Fo(T)h Ft(to)f(a)g(pair)g(\()p Fr(a;)221 b(b)p Ft(\))489 b(app)36 b(earing)489 b(in)g Fo(T)p Ft(,)503 b(w)-36 b(e)489 b(mean)g(a)g (random)800 42682 y(Fib)36 b(onacci)436 b(sequence)f(suc)-36 b(h)435 b(that)f(there)h(is)h(an)f Fr(n)h Ft(for)g(whic)-36 b(h)435 b Fr(g)31883 42881 y Fy(n)32881 42682 y Ft(=)371 b Fr(a)p Ft(,)436 b Fr(g)36367 42881 y Fy(n)p Fv(+1)38567 42682 y Ft(=)372 b Fr(b)435 b Ft(and)g Fr(n)h Ft(is)f(the)g(smallest) 800 44287 y(in)-36 b(teger)433 b(for)i(whic)-36 b(h)433 b(suc)-36 b(h)433 b(a)h(random)f(Fib)36 b(onacci)434 b(sequence)f(exists.)800 44545 y SDict begin H.S end 800 44545 a 800 44545 a SDict begin 14.5 H.A end 800 44545 a 800 44545 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.3) cvn H.B /DEST pdfmark end 800 44545 a Black 2454 x Fo(Prop)42 b(osition)581 b(3.)p Black 589 w Fs(The)530 b(r)-66 b(andom)529 b(Fib)-66 b(onac)g(ci)527 b(se)-66 b(quenc)g(e)528 b(c)-66 b(onstructe)g(d)528 b(in)g(the)i(pr)-66 b(o)g(of)528 b(of)i(Pr)-66 b(op)g(osition)p 0 .5 0 TeXcolorrgb 52136 46999 a SDict begin H.S end 52136 46999 a 0 .5 0 TeXcolorrgb Fs(1)p 0 .5 0 TeXcolorrgb 52800 46138 a SDict begin H.R end 52800 46138 a 52800 46999 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.1) cvn H.B /ANN pdfmark end 52800 46999 a Black 800 48604 a Fs(gives)464 b(the)h(only)f(shortest)i (walk)f(in)f Fo(T)h Fs(fr)-66 b(om)464 b(the)h(r)-66 b(o)g(ot)464 b(to)h(the)f(p)-66 b(air)465 b Ft(\()p Fr(a;)221 b(b)p Ft(\))p Fs(.)p Black 800 51316 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(T)-108 b(o)434 b(pro)-36 b(v)g(e)434 b(this)f(prop)36 b(osition)434 b(w)-36 b(e)433 b(need)g(the)g(follo)-36 b(wing)800 51574 y SDict begin H.S end 800 51574 a 800 51574 a SDict begin 14.5 H.A end 800 51574 a 800 51574 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.2) cvn H.B /DEST pdfmark end 800 51574 a Black 2011 x Fo(Lemma)667 b(2.)p Black 628 w Fs(L)-66 b(et)598 b(the)g(p)-66 b(air)599 b Ft(\()p Fr(a;)221 b(b)p Ft(\))599 b Fs(app)-66 b(e)g(ar)598 b(in)g Fo(T)p Fs(.)1000 b(L)-66 b(et)598 b(us)h(c)-66 b(onsider)598 b(the)h(set)f(of)h(al)66 b(l)600 b(p)-66 b(airs)598 b Ft(\()p Fr(c;)221 b(d)p Ft(\))599 b Fs(of)800 55191 y(p)-66 b(ossible)535 b(anc)-66 b(estors)535 b(of)h Ft(\()p Fr(a)347 b Ft(+)g Fr(b;)221 b(a)p Ft(\))536 b Fs(in)f Fo(T)h Fs(such)g(that)f(the)h(asc)-66 b(ending)534 b(walk)i(fr)-66 b(om)534 b Ft(\()p Fr(a;)221 b(b)p Ft(\))536 b Fs(to)f Ft(\()p Fr(c;)221 b(d)p Ft(\))536 b Fs(do)-66 b(es)800 56796 y(not)471 b(show)i(the)f(p)-66 b(air)471 b Ft(\()p Fr(a;)221 b(b)p Ft(\))p Fs(.)618 b(F)-100 b(or)472 b(al)66 b(l)473 b(such)f Ft(\()p Fr(c;)221 b(d)p Ft(\))p Fs(,)473 b(ther)-66 b(e)471 b(exist)g(four)h(inte)-66 b(gers,)471 b Fr(m)p Fs(,)h Fr(m)43389 56314 y Fw(0)43699 56796 y Fs(,)h Fr(n)g Fs(and)e Fr(n)49126 56314 y Fw(0)49437 56796 y Fs(,)i(such)800 58401 y(that)425 b Fr(c)369 b Ft(=)g Fr(ma)207 b Ft(+)g Fr(nb)425 b Fs(and)g Fr(d)369 b Ft(=)f Fr(m)16775 57919 y Fw(0)17086 58401 y Fr(a)207 b Ft(+)g Fr(n)19971 57919 y Fw(0)20281 58401 y Fr(d)p Fs(,)433 b Fq(j)p Fr(mn)24072 57919 y Fw(0)24590 58401 y Fq(\241)207 b Fr(m)26968 57919 y Fw(0)27278 58401 y Fr(n)p Fq(j)370 b Ft(=)e(1)426 b Fs(\(in)f(p)-66 b(articular,)432 b Fr(m)425 b Fs(and)g Fr(m)44831 57919 y Fw(0)45567 58401 y Fs(ar)-66 b(e)425 b(r)-66 b(elatively)800 60006 y(prime,)464 b(and)g(so)i(ar)-66 b(e)464 b Fr(n)i Fs(and)f Fr(n)15869 59524 y Fw(0)16180 60006 y Fs(\))g(and)g(such)g(that)g Fr(m)369 b(>)f(m)29337 59524 y Fw(0)30017 60006 y Ft(=)-221 b Fq(\))368 b Fr(n)h Fq(\270)h Fr(n)35828 59524 y Fw(0)36604 60006 y Fs(and)464 b Fr(n)370 b(>)e(n)42429 59524 y Fw(0)43109 60006 y Ft(=)-221 b Fq(\))369 b Fr(m)g Fq(\270)g Fr(m)49644 59524 y Fw(0)49954 60006 y Fs(.)2751 62275 y Ft(In)484 b(this)f(lemma,)497 b(the)483 b(\\p)36 b(ossible)485 b(ancestors")f(of)h(\()p Fr(a)329 b Ft(+)g Fr(b;)221 b(a)p Ft(\))484 b(are)g(the)f(pairs)h(of)g(in)-36 b(tegers)484 b(app)36 b(earing)800 63880 y(in)487 b Fo(T)h Ft(and)e(ha)-36 b(ving)488 b(\()p Fr(a)332 b Ft(+)f Fr(b;)221 b(a)p Ft(\))487 b(as)h(successors.)740 b(In)487 b(other)f(w)-36 b(ords,)501 b(these)487 b(are)h(the)e(elemen)-36 b(ts)487 b(of)h(the)f(set)800 65485 y(of)496 b(all)g(ancestors)g(of)g(all)g(pairs)g(of)g(no)36 b(des)495 b(with)h(v)-72 b(alue)495 b Fr(a)338 b Ft(+)e Fr(b)496 b Ft(at)f(paren)-36 b(t's)495 b(p)36 b(osition)496 b(and)f Fr(a)g Ft(for)h(c)-36 b(hild's)800 67090 y(p)36 b(osition)434 b(\(and,)f(so,)h Fr(b)g Ft(for)g(grandc)-36 b(hild\).)p Black 26475 70521 a(5)p Black eop %%Page: 6 6 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 1328 x Fs(Pr)-66 b(o)g(of.)p Black 649 w Ft(The)435 b(prop)36 b(ert)-36 b(y)435 b(is)g(routinely)h(v)-36 b(eri\257ed)435 b(for)g(the)g(\257rst)f(no)36 b(des)435 b(of)h(the)e(\\ascending)i (tree")f(starting)800 -1067 y(from)529 b(\()p Fr(a)360 b Ft(+)f Fr(b;)221 b(a)p Ft(\).)864 b(Let)528 b(us)g(start)h(from)g (the)f(pair)g(\()p Fr(ma)360 b Ft(+)f Fr(nb;)221 b(m)33210 -1549 y Fw(0)33521 -1067 y Fr(a)360 b Ft(+)g Fr(n)36712 -1549 y Fw(0)37023 -1067 y Fr(b)p Ft(\).)863 b(The)529 b(p)36 b(ossible)529 b(paren)-36 b(ts)527 b(of)800 539 y Fr(ma)322 b Ft(+)f Fr(nb)474 b Ft(are)f(\()p Fr(m)322 b Ft(+)f Fr(m)12723 56 y Fw(0)13034 539 y Ft(\))p Fr(a)g Ft(+)h(\()p Fr(n)g Ft(+)g Fr(n)19592 56 y Fw(0)19903 539 y Ft(\))p Fr(b)p Ft(,)483 b(for)474 b(whic)-36 b(h)473 b(the)f(required)h(prop)36 b(erties)473 b(are)g(trivially)i(v)-36 b(eri\257ed,)800 2144 y(and)433 b Fr(d)369 b Ft(:=)g Fq(j)p Ft(\()p Fr(m)295 b Fq(\241)g Fr(m)10890 1662 y Fw(0)11200 2144 y Ft(\))p Fr(a)g Ft(+)g(\()p Fr(n)g Fq(\241)g Fr(n)17672 1662 y Fw(0)17983 2144 y Ft(\))p Fr(b)p Fq(j)p Ft(.)579 b(Let)433 b(us)g(consider)g(this)h(second)f(case.)2751 3749 y(Since)322 b Fr(m)369 b(>)f(m)10061 3267 y Fw(0)10694 3749 y Ft(implies)323 b Fr(n)369 b Fq(\270)g Fr(n)18321 3267 y Fw(0)18955 3749 y Ft(and)322 b Fr(n)369 b(>)g(n)24675 3267 y Fw(0)25308 3749 y Ft(implies)323 b Fr(m)368 b Fq(\270)h Fr(m)33658 3267 y Fw(0)33969 3749 y Ft(,)345 b(w)-36 b(e)322 b(ha)-36 b(v)g(e)323 b Fr(d)369 b Ft(=)f Fq(j)p Fr(m)68 b Fq(\241)g Fr(m)45605 3267 y Fw(0)45916 3749 y Fq(j)p Fr(a)g Ft(+)g Fq(j)p Fr(n)g Fq(\241)g Fr(n)51206 3267 y Fw(0)51517 3749 y Fq(j)p Fr(b)p Ft(.)800 5354 y(In)433 b(an)-36 b(y)434 b(case,)h(the)e(prop)36 b(ert)-36 b(y)433 b Fq(j)p Fr(mn)17831 4872 y Fw(0)18437 5354 y Fq(\241)295 b Fr(m)20903 4872 y Fw(0)21213 5354 y Fr(n)p Fq(j)369 b Ft(=)g(1)434 b(is)g(v)-36 b(eri\257ed)433 b(for)h(\()p Fr(d;)221 b(ma)295 b Ft(+)g Fr(nb)p Ft(\).)2751 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23010 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.2) cvn H.B /ANN pdfmark end 52800 23010 a Black 800 24615 a Ft(is)434 b(pro)-36 b(v)g(ed.)p 51860 24615 45 878 v 51905 23781 781 45 v 51905 24615 V 52684 24615 45 878 v 2751 27216 a(Lemma)p 0 .5 0 TeXcolorrgb 7393 27216 a SDict begin H.S end 7393 27216 a 0 .5 0 TeXcolorrgb Ft(2)p 0 .5 0 TeXcolorrgb 8044 26360 a SDict begin H.R end 8044 26360 a 8044 27216 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.2) cvn H.B /ANN pdfmark end 8044 27216 a Black 433 w Ft(has)434 b(this)f(imp)36 b(ortan)-36 b(t)433 b(consequence:)800 27475 y SDict begin H.S end 800 27475 a 800 27475 a SDict begin 14.5 H.A end 800 27475 a 800 27475 a SDict begin [ /View [/XYZ H.V] /Dest (Cor.2) cvn H.B /DEST pdfmark end 800 27475 a Black 2011 x Fo(Corollary)737 b(2.)p Black 650 w(\(c)-42 b(haracterization) 719 b(of)714 b(shortest)j(w)-42 b(alks\))639 b Fs(L)-66 b(et)637 b Ft(\()p Fr(a;)221 b(b)p Ft(\))639 b Fs(b)-66 b(e)637 b(a)h(p)-66 b(air)637 b(app)-66 b(e)g(aring)637 b(in)800 31091 y Fo(T)p Fs(.)760 b(The)519 b(shortest)h(way)f(in)g Fo(T)g Fs(fr)-66 b(om)518 b(the)h(r)-66 b(o)g(ot)518 b(to)h(the)g(p)-66 b(air)519 b Ft(\()p Fr(a;)221 b(b)p Ft(\))519 b Fs(is)g(char)-66 b(acterize)g(d)518 b(by)g(the)h(fol)66 b(lowing)800 32696 y(pr)-66 b(op)g(erty:)596 b(for)465 b(any)g(p)-66 b(air)464 b Ft(\()p Fr(c;)221 b(d)p Ft(\))465 b Fs(app)-66 b(e)g(aring)463 b(in)h(this)h(walk,)g(the)f(p)-66 b(ar)g(ent)464 b(of)h Fr(c)f Fs(is)h Fq(j)p Fr(c)295 b Fq(\241)g Fr(d)p Fq(j)p Fs(.)p Black 800 35297 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(Let)430 b(\()p Fr(c;)221 b(d)p Ft(\))429 b(b)36 b(e)430 b(a)g(pair)g(app)36 b(earing)430 b(in)g Fo(T)g Ft(whic)-36 b(h)430 b(b)36 b(elongs)430 b(to)g(the)f(shortest)h(w)-36 b(alk)431 b(from)f(the)g(ro)36 b(ot)800 36902 y(to)535 b(the)e(pair)i(\()p Fr(a;)221 b(b)p Ft(\).)881 b(If)535 b(the)f(paren)-36 b(t)533 b(of)i Fr(c)g Ft(is)f Fr(c)364 b Ft(+)f Fr(d)535 b 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Ft(\)\).)634 b(The)452 b(tree)g Fo(R)g Ft(is)h(the)f Fs(r)-66 b(estricte)g(d)479 b(subtr)-66 b(e)g(e)451 b Ft(of)i Fo(T)p Ft(.)635 b(W)-108 b(e)452 b(denote)g(b)-36 b(y)452 b Fr(r)36 b Ft(\()p Fr(a;)221 b(b)p Ft(\))453 b(the)e(v)-72 b(alue)453 b(of)g(the)f(ro)-36 b(w)800 7817 y(in)433 b(whic)-36 b(h)434 b(the)f(edge)g(con)-36 b(taining)434 b Fr(a)f Ft(as)h(a)g(paren)-36 b(t)433 b(and)g Fr(b)h Ft(as)g(a)f(c)-36 b(hild)434 b(app)36 b(ears)433 b(in)h(the)f(tree)g Fo(R)p Ft(.)2751 9422 y(Lemma)p 0 .5 0 TeXcolorrgb 7417 9422 a SDict begin H.S end 7417 9422 a 0 .5 0 TeXcolorrgb Ft(3)p 0 .5 0 TeXcolorrgb 8067 8566 a SDict begin H.R end 8067 8566 a 8067 9422 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.3) cvn H.B /ANN pdfmark end 8067 9422 a Black 457 w Ft(ensure)455 b(that)i(the)f(de\257nition)g(of)h Fo(R)f Ft(is)h(non-am)-36 b(biguous,)462 b(at)457 b(a)g(single)g(exception)g(whic)-36 b(h)800 11027 y(is)434 b(treated)f(in)g(Lemma)p 0 .5 0 TeXcolorrgb 12748 11027 a SDict begin H.S end 12748 11027 a 0 .5 0 TeXcolorrgb Ft(4)p 0 .5 0 TeXcolorrgb 13398 10171 a SDict begin H.R end 13398 10171 a 13398 11027 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.4) cvn H.B /ANN pdfmark end 13398 11027 a Black Ft(.)800 11027 y SDict begin H.S end 800 11027 a 800 11027 a SDict begin 14.5 H.A end 800 11027 a 800 11027 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.3) cvn H.B /DEST pdfmark end 800 11027 a Black 2712 x Fo(Lemma)509 b(3.)p Black 558 w Fs(A)-33 b(ny)471 b(e)-66 b(dge)472 b Ft(\()p Fr(a;)221 b(b)p Ft(\))472 b Fs(app)-66 b(e)g(aring)471 b(in)h Fo(T)h Fs(app)-66 b(e)g(ars)472 b(only)h(onc)-66 b(e)471 b(in)h(the)g(r)-66 b(ow)472 b Fr(r)36 b Ft(\()p Fr(a;)221 b(b)p Ft(\))p Fs(,)475 b(ap)-66 b(art)472 b(fr)-66 b(om)800 15344 y(the)465 b(p)-66 b(air)464 b Ft(\(0)p Fr(;)221 b Ft(1\))466 b Fs(which)f(app)-66 b(e)g(ars)465 b(twic)-66 b(e)463 b(in)h(the)h(se)-66 b(c)g(ond)464 b(and)h(thir)-66 b(d)464 b(r)-66 b(ows.)p Black 800 18056 a(Pr)g(o)g(of.)p Black 649 w Ft(The)456 b(case)g(of)h(the)e(pair)h(\(0)p Fr(;)221 b Ft(1\))457 b(is)f(man)-36 b(ually)457 b(treated.)644 b(Apart)456 b(from)g(this)g(case,)462 b(w)-36 b(e)456 b(kno)-36 b(w)457 b(from)800 19661 y(the)441 b(c)-36 b(haracterization)442 b(of)g(shortest)f(w)-36 b(alks)443 b(that)e(the)g(shortest)g(w)-36 b(alk)443 b(in)e Fo(T)h Ft(from)g(the)f(ro)36 b(ot)441 b(to)h(the)f(pair)800 21266 y(\()p Fr(a;)221 b(b)p Ft(\))390 b(is)h(suc)-36 b(h)389 b(that,)399 b(for)390 b(an)-36 b(y)391 b(pair)f(\()p Fr(c;)221 b(d)p Ft(\))390 b(app)36 b(earing)390 b(in)g(this)g(w)-36 b(alk,)400 b Fr(c)p Ft('s)390 b(paren)-36 b(t)389 b(is)i Fq(j)p Fr(d)206 b Fq(\241)g Fr(c)p Fq(j)p Ft(,)399 b(so)391 b(this)f(w)-36 b(alk)800 22871 y(is)434 b(unique)f(and)g(Lemma)p 0 .5 0 TeXcolorrgb 13507 22871 a SDict begin H.S end 13507 22871 a 0 .5 0 TeXcolorrgb Ft(3)p 0 .5 0 TeXcolorrgb 14157 22015 a SDict begin H.R end 14157 22015 a 14157 22871 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.3) cvn H.B /ANN pdfmark end 14157 22871 a Black 433 w Ft(is)h(pro)-36 b(v)g(ed.)p 51860 22871 45 878 v 51905 22038 781 45 v 51905 22871 V 52684 22871 45 878 v 800 24126 a SDict begin H.S end 800 24126 a 800 24126 a SDict begin 14.5 H.A end 800 24126 a 800 24126 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.4) cvn H.B /DEST pdfmark end 800 24126 a Black 1458 x Fo(Lemma)499 b(4.)p Black 554 w Fs(The)465 b(value)g(zer)-66 b(o)464 b(app)-66 b(e)g(ars)465 b(only)g(onc)-66 b(e)464 b(in)g Fo(R)g Fs(and)h(has)h(no)f(gr)-66 b(andchild.)p Black 800 28296 a(Pr)g(o)g(of.)p Black 649 w Ft(By)538 b(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 14119 28296 a SDict begin H.S end 14119 28296 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 14769 27440 a SDict begin H.R end 14769 27440 a 14769 28296 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.1) cvn H.B /ANN pdfmark end 14769 28296 a Black Ft(,)563 b(the)537 b(v)-72 b(alue)538 b(0)f(app)36 b(ears)537 b(only)h(as)g(a)f(c)-36 b(hild)537 b(of)h(1,)564 b(so,)g(b)-36 b(y)537 b(de\257nition)f(of)i Fo(R)p Ft(,)800 29901 y(it)547 b(cannot)f(app)36 b(ears)546 b(t)-36 b(wice.)918 b(Its)546 b(c)-36 b(hildren)546 b(in)g Fo(T)h Ft(are)g(1)g(and)f(1)h (\(this)f(is)g(the)g(am)-36 b(biguit)g(y)547 b(case)g(in)g(the)800 31506 y(de\257nition)441 b(of)h Fo(R)p Ft(\),)h(whose)f(c)-36 b(hildren)440 b(in)i Fo(T)g Ft(are)f(also)i(1)e(and)g(1)h(and)f(ha)-36 b(v)g(e)442 b(to)f(b)36 b(e)441 b(exclude)h(from)g Fo(R)f Ft(since)800 33111 y(the)433 b(pair)h(\(1)p Fr(;)221 b Ft(1\))434 b(already)h(app)36 b(ears)433 b(in)h(the)f(ro)36 b(ot)434 b(of)g Fo(R)p Ft(.)p 51860 33111 45 878 v 51905 32277 781 45 v 51905 33111 V 52684 33111 45 878 v 2751 35712 a(Lemma)p 0 .5 0 TeXcolorrgb 7492 35712 a SDict begin H.S end 7492 35712 a 0 .5 0 TeXcolorrgb Ft(4)p 0 .5 0 TeXcolorrgb 8142 34856 a SDict begin H.R end 8142 34856 a 8142 35712 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.4) cvn H.B /ANN pdfmark end 8142 35712 a Black 531 w Ft(indicates)532 b(that)f(the)g(no)36 b(de)532 b(0)g(of)g Fo(R)f Ft(can)h(b)36 b(e)531 b(remo)-36 b(v)g(ed.)873 b(In)531 b(the)g(follo)-36 b(wing,)559 b(this)531 b(no)36 b(de)800 37317 y(\(and)444 b(its)h(c)-36 b(hildren\))443 b(are)i(not)f(considered)g(as)h(elemen)-36 b(ts)444 b(of)i Fo(R)p Ft(.)611 b(W)-108 b(e)444 b(will)i(call)39879 36978 y(~)39646 37317 y Fo(R)f Ft(the)f(tree)g(obtained)g(b)-36 b(y)800 38922 y(adding)453 b(to)f Fo(R)h Ft(the)f(no)36 b(de)453 b(0)g(corresp)36 b(onding)452 b(to)h(the)f(left)i(c)-36 b(hild)452 b(of)i(the)e(second)g(1.)637 b(The)452 b(tree)47652 38583 y(~)47420 38922 y Fo(R)g Ft(will)i(b)36 b(e)800 40527 y(useful)434 b(to)f(in)-36 b(v)g(estigate)435 b(the)e(p)36 b(ositions)434 b(of)g(the)f(0)h(no)36 b(des)433 b(in)h Fo(T)g Ft(\(see)f(section)p 0 .5 0 TeXcolorrgb 38706 40527 a SDict begin H.S end 38706 40527 a 0 .5 0 TeXcolorrgb Ft(4.2)p 0 .5 0 TeXcolorrgb 40368 39671 a SDict begin H.R end 40368 39671 a 40368 40527 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.4.2) cvn H.B /ANN pdfmark end 40368 40527 a Black Ft(\).)2751 42133 y(Here)593 b(are)f(the)g(\257rst) g(few)h(ro)-36 b(ws)593 b(of)g Fo(R)p Ft(.)1055 b(As)593 b(in)f(the)g(case)h(of)g Fo(T)p Ft(,)633 b(a)593 b(left)g(c)-36 b(hild)592 b(corresp)36 b(onds)592 b(to)h(a)800 43738 y(subtraction)552 b(and)h(a)g(righ)-36 b(t)553 b(c)-36 b(hild)553 b(to)g(an)g(addition.)937 b(When)553 b(a)g(no)36 b(de)553 b(has)g(only)h(one)f(c)-36 b(hild,)583 b(w)-36 b(e)553 b(use)g(a)800 45343 y(v)-36 b(ertical)365 b(line;)388 b(Lemma)p 0 .5 0 TeXcolorrgb 12700 45343 a SDict begin H.S end 12700 45343 a 0 .5 0 TeXcolorrgb Ft(5)p 0 .5 0 TeXcolorrgb 13350 44487 a SDict begin H.R end 13350 44487 a 13350 45343 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.5) cvn H.B /ANN pdfmark end 13350 45343 a Black 364 w Ft(will)365 b(sho)-36 b(w)365 b(that)e(suc)-36 b(h)363 b(a)i(v)-36 b(ertical)365 b(edge)f(alw)-36 b(a)g(ys)365 b(corresp)36 b(onds)364 b(to)g(an)g(addition.)p Black 26475 70521 a(7)p Black eop %%Page: 8 8 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black -3024 0 a @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 9.5 9.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3150 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 8.5 8.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3149 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 7.5 7.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3150 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 6.5 6.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3149 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 5.5 5.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3150 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 4.5 4.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3150 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 3.5 3.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3149 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 2.5 2.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3150 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 1.5 1.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 3149 x @beginspecial @setspecial 18 -1 sub 1864679 mul 65536 div 72.27 div 72 mul 0.5 0.5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 3 3 ] psline @endspecial 20972 -1575 a @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 9 10 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 x @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 8 9 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3149 x @beginspecial @setspecial 2.455 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 7 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.7825 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 7 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 7858 7874 a @beginspecial @setspecial 2.455 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 26229 w @beginspecial @setspecial 8.24 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.325 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 7858 11024 a @beginspecial @setspecial 0.86 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.05 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 18221 w @beginspecial @setspecial 8.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 16015 w @beginspecial @setspecial 11.74 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.91 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2835 14173 a @beginspecial @setspecial 0.86 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 10047 w @beginspecial @setspecial 3.08 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.02 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 13197 w @beginspecial @setspecial 7.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.18 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 11023 w @beginspecial @setspecial 11.74 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 9985 w @beginspecial @setspecial 13.92 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.9 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2835 17323 a @beginspecial @setspecial 0.24 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.48 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 3.08 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 4.4 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.64 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 7.24 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 8.56 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.8 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8063 w @beginspecial @setspecial 11.12 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.36 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6866 w @beginspecial @setspecial 13.92 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6237 w @beginspecial @setspecial 15.28 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.52 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 882 20472 a @beginspecial @setspecial 0.24 0.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 1.12 1.48 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.84 1.48 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 5040 w @beginspecial @setspecial 2.72 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 3.44 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4157 w @beginspecial @setspecial 4.4 4.4 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3906 w @beginspecial @setspecial 5.28 5.64 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 6 5.64 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 5039 w @beginspecial @setspecial 6.88 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 7.6 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4158 w @beginspecial @setspecial 8.56 8.56 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 9.44 9.8 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.16 9.8 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4158 w @beginspecial @setspecial 11.12 11.12 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 12 12.36 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.72 12.36 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4913 w @beginspecial @setspecial 13.6 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.32 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4284 w @beginspecial @setspecial 15.28 15.28 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 16.16 16.52 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.88 16.52 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 882 23622 a @beginspecial @setspecial 0 0.24 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 0.48 0.24 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2772 w @beginspecial @setspecial 1.12 1.12 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2267 w @beginspecial @setspecial 1.6 1.84 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 2.08 1.84 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2772 w @beginspecial @setspecial 2.72 2.72 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2268 w @beginspecial @setspecial 3.2 3.44 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 3.68 3.44 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3023 w @beginspecial @setspecial 4.16 4.4 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.64 4.4 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2772 w @beginspecial @setspecial 5.28 5.28 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2268 w @beginspecial @setspecial 5.76 6 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 6.24 6 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2771 w @beginspecial @setspecial 6.88 6.88 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2268 w @beginspecial @setspecial 7.36 7.6 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 7.84 7.6 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3024 w @beginspecial @setspecial 8.32 8.56 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 8.8 8.56 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2771 w @beginspecial @setspecial 9.44 9.44 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2268 w @beginspecial @setspecial 9.92 10.16 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.4 10.16 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3024 w @beginspecial @setspecial 10.88 11.12 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 11.36 11.12 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2771 w @beginspecial @setspecial 12 12 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2268 w @beginspecial @setspecial 12.48 12.72 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.96 12.72 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2772 w @beginspecial @setspecial 13.6 13.6 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2267 w @beginspecial @setspecial 14.08 14.32 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.56 14.32 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3024 w @beginspecial @setspecial 15.04 15.28 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.52 15.28 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2772 w @beginspecial @setspecial 16.16 16.16 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2267 w @beginspecial @setspecial 16.64 16.88 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 17.12 16.88 sub 1864679 mul 65536 div 72.27 div 72 mul 1 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 126 26772 a @beginspecial @setspecial 0 0 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 0.32 0.48 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 0.64 0.48 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2016 w @beginspecial @setspecial 0.96 1.12 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.28 1.12 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1511 w @beginspecial @setspecial 1.6 1.6 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 1.92 2.08 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 2.24 2.08 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2016 w @beginspecial @setspecial 2.56 2.72 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 2.88 2.72 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 3.2 3.2 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 3.52 3.68 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 3.84 3.68 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1511 w @beginspecial @setspecial 4.16 4.16 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 4.48 4.64 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.8 4.64 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2016 w @beginspecial @setspecial 5.12 5.28 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.44 5.28 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 5.76 5.76 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 6.08 6.24 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 6.4 6.24 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2015 w @beginspecial @setspecial 6.72 6.88 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 7.04 6.88 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 7.36 7.36 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 7.68 7.84 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 8 7.84 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 8.32 8.32 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 8.64 8.8 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 8.96 8.8 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2015 w @beginspecial @setspecial 9.28 9.44 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.6 9.44 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 9.92 9.92 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 10.24 10.4 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.56 10.4 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 10.88 10.88 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 11.2 11.36 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 11.52 11.36 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2015 w @beginspecial @setspecial 11.84 12 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.16 12 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 12.48 12.48 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 12.8 12.96 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.12 12.96 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2016 w @beginspecial @setspecial 13.44 13.6 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.76 13.6 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1511 w @beginspecial @setspecial 14.08 14.08 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 14.4 14.56 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.72 14.56 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 15.04 15.04 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 15.36 15.52 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.68 15.52 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2016 w @beginspecial @setspecial 16 16.16 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.32 16.16 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1511 w @beginspecial @setspecial 16.64 16.64 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1512 w @beginspecial @setspecial 16.96 17.12 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 17.28 17.12 sub 1864679 mul 65536 div 72.27 div 72 mul 0 1 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 20972 -1575 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 x @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3149 x @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 7858 7874 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 26229 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 7858 11024 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 18221 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 16015 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2835 14173 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 10047 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 13197 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 11023 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 9985 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2835 17323 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6992 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6110 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6992 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6110 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 8063 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6866 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6237 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 882 20472 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3905 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 5040 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4157 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3906 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 5039 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4158 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3905 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4158 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3905 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4913 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4284 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3905 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 882 23622 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2772 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2267 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2772 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2268 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3023 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2772 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2268 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2771 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2268 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3024 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2771 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2268 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3024 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2771 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2268 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2772 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2267 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3024 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2772 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2267 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 126 26772 a @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2016 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1511 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2016 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1511 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2016 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2015 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2015 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2015 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2016 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1511 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 2016 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1511 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1512 w @beginspecial @setspecial 0.552 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 126 29921 a @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1007 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1007 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1007 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1007 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1007 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1007 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.268 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1007 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1008 w @beginspecial @setspecial 0.397 1864679 mul 65536 div 72.27 div 72 mul 0.275 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 0 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 3150 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 6299 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 9449 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 12598 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 15748 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 18898 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 22047 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 25197 a @beginspecial @setspecial 0.519 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial -3024 28346 a @beginspecial @setspecial 0.669 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 59843 w @beginspecial @setspecial 0.139 1864679 mul 65536 div 72.27 div 72 mul 0.139 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 20647 -1147 a Ft(1)20647 2003 y(1)20647 5152 y(2)7533 8302 y(1)25578 b(3)7533 11452 y(3)17571 b(1)15365 b(5)2509 14601 y(2)9398 b(4)12547 b(4)10373 b(2)9334 b(8)2509 17751 y(5)6343 b(1)5460 b(7)6342 b(3)5460 b(5)7413 b(7)6216 b(3)5261 b(13)557 20900 y(3)3255 b(7)4390 b(5)3507 b(3)2930 b(11)4065 b(7)3507 b(1)3256 b(9)3507 b(5)3256 b(9)3938 b(11)3309 b(5)2930 b(21)557 24050 y(8)2121 b(2)1293 b(12)1797 b(4)1617 b(6)2049 b(10)1797 b(4)1292 b(18)1797 b(4)1293 b(10)2048 b(6)2122 b(4)1293 b(14)1723 b(12)1797 b(2)1293 b(16)1796 b(8)1293 b(14)1724 b(18)1796 b(8)1293 b(34)-199 27200 y(5)536 b(11)1041 b(9)862 b(5)537 b(19)1041 b(9)862 b(1)536 b(11)h(7)g(13)716 b(15)537 b(7)f(29)716 b(11)537 b(3)g(17)g(5)861 b(7)1041 b(13)537 b(5)g(23)g(7)f(17)716 b(11)537 b(7)g(25)715 b(19)537 b(3)g(25)212 b(13)g(23)715 b(29)212 b(13)g(55)-280 30135 y Fl(13)399 b(3)g(19)g(7)g(11)195 b(17)399 b(7)g(31)g(5)g(13)g(7)602 b(5)399 b(17)195 b(17)399 b(3)g(23)196 b(11)g(19)g(25)g(11)g(47)398 b(7)h(15)196 b(13)399 b(7)g(27)196 b(11)399 b(1)g(13)f(9)h(17)196 b(19)399 b(9)g(37)196 b(19)399 b(5)g(29)f(9)h(13)196 b(23)399 b(9)g(41)196 b(11)g(27)g(17)f(11)h(39)g(31)399 b(5)g(41)196 b(21)g(37)g(47)f(21)h(89)-3622 157 y Fr(\275)-2951 356 y Fv(1)-3622 3306 y Fr(\275)-2951 3505 y Fv(2)-3622 6456 y Fr(\275)-2951 6655 y Fv(3)-3622 9606 y Fr(\275)-2951 9805 y Fv(4)-3622 12755 y Fr(\275)-2951 12954 y Fv(5)-3622 15905 y Fr(\275)-2951 16104 y Fv(6)-3622 19054 y Fr(\275)-2951 19253 y Fv(7)-3622 22204 y Fr(\275)-2951 22403 y Fv(8)-3622 25354 y Fr(\275)-2951 25553 y Fv(9)-3857 28503 y Fr(\275)-3186 28702 y Fv(10)800 31496 y SDict begin H.S end 800 31496 a 800 31496 a SDict begin 14.5 H.A end 800 31496 a 800 31496 a SDict begin [ /View [/XYZ H.V] /Dest (figure.2) cvn H.B /DEST pdfmark end 800 31496 a Black 15223 34208 a Ft(Figure)433 b(2:)p 0 TeXcolorgray 20913 32603 a SDict begin H.S end 20913 32603 a 0 TeXcolorgray 0 TeXcolorgray 20913 32603 a SDict begin H.R end 20913 32603 a 20913 32603 a SDict begin [ /View [/XYZ H.V] /Dest (figure.2) cvn H.B /DEST pdfmark end 20913 32603 a Black 1605 x Ft(The)g(restricted)g(tree)h Fo(R)368 b Ft(=)h Fo(R)35955 34415 y Fv(\(1)p Fy(;)p Fv(1\))p Black Black 2751 37578 a Ft(Let)423 b(us)g(remark,)j(again)e (without)g(further)e(elab)36 b(oration,)427 b(that)c(the)g(sequence)g (of)h(lab)36 b(els)424 b(in)f(the)g(tree)800 39183 y Fo(R)p Ft(,)377 b(read)362 b(in)h(breadth-\257rst)d(order)j(\(1,)377 b(1,)g(2,)h(1,)f(3,)h(3,)f(1,)h(5,)f(2,)h(4,)f(4,)h(2,)f(8.)217 b(.)g(.)g(\))555 b(is)363 b(a)g Fr(\257)74 b Fs(-r)-66 b(e)g(gular)399 b(se)-66 b(quenc)g(e)p Ft(,)376 b(as)800 40789 y(de\257ned)382 b(b)-36 b(y)384 b(Allouc)-36 b(he,)395 b(Sc)-36 b(heic)g(her)382 b(and)i(Tic)-36 b(h)g(y)384 b([)p 0 .5 0 TeXcolorrgb 24754 40789 a SDict begin H.S end 24754 40789 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 25405 39932 a SDict begin H.R end 25405 39932 a 25405 40789 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.AST) cvn H.B /ANN pdfmark end 25405 40789 a Black Ft(],)394 b(where)384 b(here)g Fr(\257)458 b Ft(is)384 b(the)f(n)-36 b(umeration)383 b(system)i(de\257ned)800 42394 y(b)-36 b(y)434 b(the)f(Fib)36 b(onacci)434 b(sequence.)800 42652 y SDict begin H.S end 800 42652 a 800 42652 a SDict begin 14.5 H.A end 800 42652 a 800 42652 a SDict begin [ /View [/XYZ H.V] /Dest (Notation.1) cvn H.B /DEST pdfmark end 800 42652 a Black 2135 x Fo(Notation)491 b(1.)p Black 548 w Fs(L)-66 b(et)456 b Fr(a)i Fs(b)-66 b(e)457 b(a)h(no)-66 b(de)456 b(of)i Fo(R)p Fs(.)594 b(If)457 b Fr(a)h Fs(has)g(a)g(left)f (\(r)-66 b(esp.)595 b(right\))457 b(child,)h(it)f(is)h(denote)-66 b(d)456 b(by)i Fr(c)50319 44305 y Fw(\241)51106 44787 y Ft(\()p Fr(a)p Ft(\))800 46392 y Fs(\(r)-66 b(esp.)597 b Fr(c)5162 45910 y Fv(+)5949 46392 y Ft(\()p Fr(a)p Ft(\))p Fs(\).)800 47512 y SDict begin H.S end 800 47512 a 800 47512 a SDict begin 14.5 H.A end 800 47512 a 800 47512 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.5) cvn H.B /DEST pdfmark end 800 47512 a Black 1273 x Fo(Lemma)499 b(5.)p Black 554 w Fs(In)464 b Fo(R)p Fs(,)g(if)g Fr(b)h Fs(is)g(the)f(left)g(child)h(of)g Fr(a)p Fs(,)f(then)g Fr(b)h Fs(has)h(no)e(left)g(child.)p Black 800 51178 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(Let)311 b(assume)h(that)f(w)-36 b(e)311 b(can)h(\257nd)e(three)h(no)36 b(des)311 b Fr(x)p Ft(,)336 b Fr(y)359 b Ft(and)311 b Fr(z)371 b Ft(suc)-36 b(h)311 b(that)g Fr(z)428 b Ft(=)369 b Fr(c)41711 50696 y Fw(\241)42498 51178 y Ft(\()p Fr(y)48 b Ft(\))311 b(and)g Fr(y)416 b Ft(=)369 b Fr(c)49901 50696 y Fw(\241)50688 51178 y Ft(\()p Fr(x)p Ft(\).)800 52783 y(By)434 b(considering)g (successiv)-36 b(e)434 b(paren)-36 b(ts)433 b(if)h(necessary)-108 b(,)434 b(w)-36 b(e)434 b(can)f(supp)36 b(ose)433 b(that)g Fr(x)369 b Ft(=)g Fr(c)42518 52301 y Fv(+)43305 52783 y Ft(\()p Fr(w)36 b Ft(\).)2751 54388 y(Let)388 b Fr(v)437 b Ft(b)36 b(e)388 b(the)g(paren)-36 b(t)388 b(of)i Fr(w)36 b Ft(.)563 b(Then,)397 b(w)-36 b(e)389 b(ha)-36 b(v)g(e)389 b Fr(x)370 b Ft(=)e Fr(v)251 b Ft(+)204 b Fr(w)36 b Ft(,)397 b(so)389 b Fr(y)417 b Ft(=)369 b Fq(j)p Fr(x)204 b Fq(\241)g Fr(w)36 b Fq(j)368 b Ft(=)g Fr(v)437 b Ft(and)388 b Fr(z)428 b Ft(=)369 b Fq(j)p Fr(y)251 b Fq(\241)204 b Fr(x)p Fq(j)369 b Ft(=)800 55993 y Fq(j)p Fr(v)301 b Fq(\241)253 b Ft(\()p Fr(v)301 b Ft(+)253 b Fr(w)36 b Ft(\))p Fq(j)368 b Ft(=)h Fr(w)36 b Ft(,)417 b(so)c(\()p Fr(y)48 b(;)221 b(z)59 b Ft(\))370 b(=)e(\()p Fr(v)48 b(;)221 b(w)36 b Ft(\),)417 b(whic)-36 b(h)413 b(con)-36 b(tradicts)412 b(the)h(non-redundance)d (in)j(the)g(de\257nition)800 57598 y(of)434 b Fo(R)p Ft(.)p 51860 57598 45 878 v 51905 56765 781 45 v 51905 57598 V 52684 57598 45 878 v 800 58488 a SDict begin H.S end 800 58488 a 800 58488 a SDict begin 14.5 H.A end 800 58488 a 800 58488 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.6) cvn H.B /DEST pdfmark end 800 58488 a Black 1504 x Fo(Lemma)h(6.)p Black 511 w Fs(In)413 b Fo(R)p Fs(,)423 b(if)413 b Fr(a)h Fs(has)g Fr(b)g Fs(as)g(a)g(right)f(\(r)-66 b(esp.)580 b(left\))413 b(child,)424 b(then)413 b Fr(b)g Fs(is)h(smal)66 b(ler)414 b(\(r)-66 b(esp.)581 b(bigger\))412 b(than)800 61597 y Fr(a)p Fs(.)597 b(The)465 b(only)g(c)-66 b(ase)464 b(of)h(e)-66 b(quality)464 b(c)-66 b(orr)g(esp)g(onds)464 b(to)h(the)f(p)-66 b(air)465 b Ft(\(1)p Fr(;)221 b Ft(1\))465 b Fs(in)f(the)h(b)-66 b(e)g(ginning)461 b(of)k(the)f(tr)-66 b(e)g(e.)p Black 800 63990 a(Pr)g(o)g(of.)p Black 649 w Ft(Let)446 b(us)h(denote)f(b)-36 b(y)447 b Fr(z)506 b Ft(the)446 b(paren)-36 b(t)446 b(of)h Fr(a)p Ft(.)618 b(If)447 b Fr(b)391 b Ft(=)g Fr(c)29757 63508 y Fv(+)30544 63990 y Ft(\()p Fr(a)p Ft(\),)450 b(then)c(it)g(is)h(ob)-36 b(vious)448 b(that)e Fr(b)391 b(>)g(a)447 b Ft(since,)800 65595 y(apart)476 b(for)i(no)36 b(des)476 b(in)g(the)g(top)h(of)g(the)f (tree,)488 b(w)-36 b(e)476 b(ha)-36 b(v)g(e)477 b Fr(b)442 b Ft(=)g Fr(a)324 b Ft(+)h Fr(z)535 b Ft(and)477 b Fr(z)501 b(>)442 b Ft(0)477 b(b)-36 b(y)476 b(Lemma)p 0 .5 0 TeXcolorrgb 46789 65595 a SDict begin H.S end 46789 65595 a 0 .5 0 TeXcolorrgb Ft(4)p 0 .5 0 TeXcolorrgb 47439 64739 a SDict begin H.R end 47439 64739 a 47439 65595 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.4) cvn H.B /ANN pdfmark end 47439 65595 a Black 477 w Ft(\(and)g(the)800 67200 y(exclusion)434 b(of)h(the)e(0-no)36 b(de)433 b(w)-36 b(e)434 b(made)f(after)h(this)f(lemma\).)p Black 26475 70521 a(8)p Black eop %%Page: 9 9 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 2751 -2672 a Ft(If)449 b Fr(b)394 b Ft(=)f Fr(c)6978 -3154 y Fw(\241)7765 -2672 y Ft(\()p Fr(a)p Ft(\),)452 b(then)447 b(\(if)i Fr(a)f Ft(is)g(not)g(the)g(ro)36 b(ot)448 b(of)h Fo(R)p Ft(\),)j(b)-36 b(y)448 b(Lemma)p 0 .5 0 TeXcolorrgb 35202 -2672 a SDict begin H.S end 35202 -2672 a 0 .5 0 TeXcolorrgb Ft(5)p 0 .5 0 TeXcolorrgb 35853 -3528 a SDict begin H.R end 35853 -3528 a 35853 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.5) cvn H.B /ANN pdfmark end 35853 -2672 a Black 448 w Ft(w)-36 b(e)448 b(ha)-36 b(v)g(e)449 b Fr(a)393 b Ft(=)h Fr(c)44286 -3154 y Fv(+)45073 -2672 y Ft(\()p Fr(z)59 b Ft(\))448 b(so,)k(b)-36 b(y)449 b(the)800 -1067 y(b)36 b(eginning)433 b(of)i(the)e(pro)36 b(of,)434 b Fr(a)369 b(>)g(z)59 b Ft(,)434 b(so)g Fr(b)369 b Ft(=)f Fr(a)295 b Fq(\241)h Fr(z)428 b(<)369 b(a)p Ft(,)433 b(and)g(the)g(pro)36 b(of)435 b(is)e(complete.)p 51860 -1067 45 878 v 51905 -1900 781 45 v 51905 -1067 V 52684 -1067 45 878 v 800 188 a SDict begin H.S end 800 188 a 800 188 a SDict begin 14.5 H.A end 800 188 a 800 188 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.7) cvn H.B /DEST pdfmark end 800 188 a Black 1457 x Fo(Lemma)499 b(7.)p Black 554 w Fs(In)464 b Fo(R)p Fs(,)g(every)g(no)-66 b(de)464 b(has)i(a)f(right)f(child.)p Black 800 4358 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(Let)469 b(us)g(consider)f(the)h(no) 36 b(de)469 b Fr(b)p Ft(,)478 b(whic)-36 b(h)469 b(paren)-36 b(t)468 b(is)i Fr(a)p Ft(.)685 b(In)468 b Fo(T)p Ft(,)479 b(the)469 b(righ)-36 b(t)469 b(c)-36 b(hild)468 b(of)i Fr(b)f Ft(is)h Fr(a)319 b Ft(+)g Fr(b)p Ft(.)685 b(It)800 5963 y(remains)442 b(to)h(sho)-36 b(w)442 b(that)g(the)g(shortest)g(w) -36 b(alk)443 b(in)f Fo(T)h Ft(from)g(the)e(ro)36 b(ot)443 b(to)g(\()p Fr(b;)221 b(a)301 b Ft(+)g Fr(b)p Ft(\))442 b(is)g(suc)-36 b(h)442 b(that)g Fr(b)g Ft(has)g Fr(a)800 7568 y Ft(as)456 b(paren)-36 b(t.)645 b(But)456 b(this)g(is)g(a)h (simple)f(consequence)g(of)h(the)e(c)-36 b(haracteristic)457 b(prop)36 b(ert)-36 b(y)455 b(of)i(shortest)f(w)-36 b(alks)800 9173 y(\(Corollary)p 0 .5 0 TeXcolorrgb 7050 9173 a SDict begin H.S end 7050 9173 a 0 .5 0 TeXcolorrgb Ft(2)p 0 .5 0 TeXcolorrgb 7701 8317 a SDict begin H.R end 7701 8317 a 7701 9173 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Cor.2) cvn H.B /ANN pdfmark end 7701 9173 a Black Ft(\):)578 b(the)433 b(paren)-36 b(t)433 b(of)h Fr(b)g Ft(is)f Fq(j)p Ft(\()p Fr(a)295 b Ft(+)g Fr(b)p Ft(\))g Fq(\241)g Fr(b)p Fq(j)369 b Ft(=)g Fr(a)p Ft(.)p 51860 10778 45 878 v 51905 9944 781 45 v 51905 10778 V 52684 10778 45 878 v 800 11774 a SDict begin H.S end 800 11774 a 800 11774 a SDict begin 14.5 H.A end 800 11774 a 800 11774 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.8) cvn H.B /DEST pdfmark end 800 11774 a Black 1716 x Fo(Lemma)499 b(8.)p Black 554 w Fs(In)464 b Fo(R)p Fs(,)g(if)g(the)h(no)-66 b(de)464 b Fr(a)g Fs(has)i Fr(b)f Fs(as)h(only)e(child,)g(then)h Fr(b)g Fs(has)g(two)g(childr)-66 b(en.)p Black 800 16202 a(Pr)g(o)g(of.)p Black 649 w Ft(By)385 b(Lemma)p 0 .5 0 TeXcolorrgb 11456 16202 a SDict begin H.S end 11456 16202 a 0 .5 0 TeXcolorrgb Ft(7)p 0 .5 0 TeXcolorrgb 12107 15346 a SDict begin H.R end 12107 15346 a 12107 16202 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.7) cvn H.B /ANN pdfmark end 12107 16202 a Black Ft(,)394 b Fr(b)369 b Ft(=)f Fr(c)15724 15720 y Fv(+)16511 16202 y Ft(\()p Fr(a)p Ft(\).)562 b(By)384 b(Lemma)p 0 .5 0 TeXcolorrgb 25713 16202 a SDict begin H.S end 25713 16202 a 0 .5 0 TeXcolorrgb Ft(7)p 0 .5 0 TeXcolorrgb 26363 15346 a SDict begin H.R end 26363 15346 a 26363 16202 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.7) cvn H.B /ANN pdfmark end 26363 16202 a Black 384 w Ft(again,)395 b Fr(b)384 b Ft(has)g(a)g(righ)-36 b(t)383 b(c)-36 b(hild,)394 b(whic)-36 b(h)384 b(is)g Fr(a)194 b Ft(+)g Fr(b)p Ft(.)561 b(In)383 b Fo(T)p Ft(,)800 17807 y(the)409 b(left)h(c)-36 b(hild)409 b(of)i Fr(b)e Ft(is)h Fq(j)p Fr(a)246 b Fq(\241)g Fr(b)p Fq(j)p Ft(,)415 b(whic)-36 b(h)409 b(is)h(equal)g(to)g Fr(b)246 b Fq(\241)g Fr(a)410 b Ft(b)-36 b(y)409 b(Lemma)p 0 .5 0 TeXcolorrgb 35892 17807 a SDict begin H.S end 35892 17807 a 0 .5 0 TeXcolorrgb Ft(6)p 0 .5 0 TeXcolorrgb 36543 16951 a SDict begin H.R end 36543 16951 a 36543 17807 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.6) cvn H.B /ANN pdfmark end 36543 17807 a Black Ft(.)570 b(Again)410 b(b)-36 b(y)410 b(the)f(c)-36 b(haracteristic)800 19412 y(prop)36 b(ert)-36 b(y)357 b(of)h(shortest)f(w)-36 b(alks,)375 b(the)357 b(paren)-36 b(t)356 b(of)j Fr(b)e Ft(in)g(the)g(shortest)g(w) -36 b(alk)359 b(to)f(\()p Fr(b;)221 b(b)140 b Fq(\241)g Fr(a)p Ft(\))357 b(is)h Fq(j)p Fr(b)140 b Fq(\241)g Ft(\()p Fr(b)g Fq(\241)g Fr(a)p Ft(\))p Fq(j)368 b Ft(=)h Fr(a)p Ft(,)800 21017 y(and)433 b(w)-36 b(e)434 b(are)g(done.)p 51860 21017 45 878 v 51905 20184 781 45 v 51905 21017 V 52684 21017 45 878 v 800 22014 a SDict begin H.S end 800 22014 a 800 22014 a SDict begin 14.5 H.A end 800 22014 a 800 22014 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.9) cvn H.B /DEST pdfmark end 800 22014 a Black 1715 x Fo(Lemma)499 b(9.)p Black 554 w Fs(In)464 b Fo(R)p Fs(,)g(if)g(the)h(no)-66 b(de)464 b Fr(b)h Fs(is)f(a)i(right)e(child,)g(then)g(it)g(has)i(two)f (childr)-66 b(en.)2751 26441 y Ft(The)434 b(pro)36 b(of)434 b(go)36 b(es)434 b(as)g(in)g(Lemma)p 0 .5 0 TeXcolorrgb 19439 26441 a SDict begin H.S end 19439 26441 a 0 .5 0 TeXcolorrgb Ft(8)p 0 .5 0 TeXcolorrgb 20089 25585 a SDict begin H.R end 20089 25585 a 20089 26441 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.8) cvn H.B /ANN pdfmark end 20089 26441 a Black Ft(.)800 26700 y SDict begin H.S end 800 26700 a 800 26700 a SDict begin 14.5 H.A end 800 26700 a 800 26700 a SDict begin [ /View [/XYZ H.V] /Dest (Notation.2) cvn H.B /DEST pdfmark end 800 26700 a Black 2453 x Fo(Notation)619 b(2.)p Black 607 w Fs(The)560 b(set)g(of)g(e)-66 b(dges)559 b(fr)-66 b(om)559 b(the)h Ft(\()p Fr(n)366 b Fq(\241)g Ft(1\))p Fs(-th)560 b(to)g(the)g Fr(n)p Fs(-th)g(r)-66 b(ow)560 b(of)g Fo(R)g Fs(is)g(denote)-66 b(d)559 b(by)h Fr(\275)52174 29352 y Fy(n)800 30759 y Fs(\(by)484 b(c)-66 b(onvention,)485 b(the)f(r)-66 b(o)g(ot)483 b(de\257nes)g(the)g Ft(0)p Fs(-th)h(r)-66 b(ow,)488 b(so)d(the)e(e)-66 b(dge)483 b(fr)-66 b(om)483 b(the)g(r)-66 b(o)g(ot)483 b Ft(1)h Fs(to)g(its)g(only)f(child)h Ft(1)800 32364 y Fs(c)-66 b(orr)g(esp)g(onds)486 b(to)h Fr(\275)10083 32563 y Fv(1)10609 32364 y Fs(,)492 b(the)487 b(next)e(one)i(fr)-66 b(om)486 b Ft(1)h Fs(to)g Ft(2)h Fs(c)-66 b(or)g(esp)g(onds)486 b(to)h Fr(\275)34692 32563 y Fv(2)35217 32364 y Fs(,)492 b(and)487 b(so)h(on\).)664 b(In)486 b Fr(\275)45715 32563 y Fy(n)46341 32364 y Fs(,)492 b(the)487 b(subset)800 33969 y(of)421 b(left)e(\(r)-66 b(esp.)583 b(right\))420 b(e)-66 b(dges)421 b(is)f(denote)-66 b(d)420 b(by)g Fr(\275)23563 33487 y Fw(\241)23563 34297 y Fy(n)24771 33969 y Fs(\(r)-66 b(esp.)583 b Fr(\275)29230 33487 y Fv(+)30017 33969 y Ft(\()p Fr(n)p Ft(\))p Fs(\).)g(A)420 b(set)h Fr(X)525 b Fs(of)421 b(e)-66 b(dges)420 b(in)g Fo(R)g Fs(b)-66 b(eing)419 b(given,)800 35574 y(we)460 b(denote)e(by)h Fr(c)p Ft(\()p Fr(X)104 b Ft(\))p Fs(,)460 b Fr(c)12728 35092 y Fw(\241)13515 35574 y Ft(\()p Fr(X)104 b Ft(\))460 b Fs(and)f Fr(c)19248 35092 y Fv(+)20035 35574 y Ft(\()p Fr(X)104 b Ft(\))460 b Fs(r)-66 b(esp)g(e)g(ctively)457 b(the)i(set)g(of)g(childr)-66 b(en,)458 b(left)h(childr)-66 b(en)458 b(and)h(right)800 37179 y(childr)-66 b(en)472 b(of)h Fr(X)578 b Fs(in)472 b Fo(R)p Fs(.)622 b(We)473 b(also)h(denote)e(by)h Fr(S)77 b Ft(\()p Fr(X)104 b Ft(\))475 b Fs(the)d(sum)i(of)f(the)g(values)h(of)f(al)66 b(l)474 b(the)f(\257nal)f(no)-66 b(des)473 b(of)800 38784 y Fr(X)104 b Fs(.)2751 40389 y(We)469 b(denote)f(by)h Ft(\()p Fr(F)12317 40588 y Fy(n)12943 40389 y Ft(\))13449 40588 y Fy(n)14544 40389 y Fs(the)g(classic)-66 b(al)469 b(Fib)-66 b(onac)g(ci)466 b(se)-66 b(quenc)g(e:)604 b Fr(F)34254 40588 y Fv(0)35157 40389 y Ft(=)376 b(0)p Fs(,)470 b Fr(F)38906 40588 y Fv(1)39809 40389 y Ft(=)376 b(1)470 b Fs(and,)f(for)g(al)66 b(l)470 b Fr(n)377 b Fq(\270)g Ft(2)p Fs(,)800 41994 y Fr(F)1642 42193 y Fy(n)2637 41994 y Ft(=)369 b Fr(F)4860 42193 y Fy(n)p Fw(\241)p Fv(1)6984 41994 y Ft(+)294 b Fr(F)9132 42193 y Fy(n)p Fw(\241)p Fv(2)10961 41994 y Fs(.)2751 44706 y Ft(W)-108 b(e)434 b(can)f(no)-36 b(w)434 b(giv)-36 b(e)434 b(t)-36 b(w)g(o)434 b(quite)g(unexp)36 b(ected)433 b(prop)36 b(erties)433 b(of)h Fo(R)p Ft(.)800 44965 y SDict begin H.S end 800 44965 a 800 44965 a SDict begin 14.5 H.A end 800 44965 a 800 44965 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.4) cvn H.B /DEST pdfmark end 800 44965 a Black 2453 x Fo(Prop)42 b(osition)500 b(4.)p Black 554 w Fs(F)-100 b(or)465 b(al)66 b(l)466 b Fr(n)369 b Fq(\270)g Ft(2)p Fs(,)465 b(we)g(have)13225 50628 y Ft(Card\()p Fr(\275)17220 50080 y Fw(\241)17220 50957 y Fy(n)18007 50628 y Ft(\))369 b(=)g Fr(F)21105 50827 y Fy(n)p Fw(\241)p Fv(2)25590 50628 y Fs(and)2657 b Ft(Card\()p Fr(\275)34301 50080 y Fv(+)34301 50957 y Fy(n)35088 50628 y Ft(\))368 b(=)h Fr(F)38185 50827 y Fy(n)p Fw(\241)p Fv(1)40014 50628 y Fr(;)800 52953 y Fs(and)465 b(so)22447 54558 y Ft(Card\()p Fr(\275)26442 54757 y Fy(n)27068 54558 y Ft(\))369 b(=)g Fr(F)30166 54757 y Fy(n)30792 54558 y Fr(:)p Black 800 57270 a Fs(Pr)-66 b(o)g(of.)p Black 649 w Ft(The)382 b(prop)36 b(ert)-36 b(y)382 b(is)h(easily)h(v) -36 b(eri\257ed)382 b(for)h(the)e(\257rst)h(ro)-36 b(ws.)562 b(Let)381 b(assume)i(that)f(the)f(prop)36 b(ert)-36 b(y)382 b(is)h(true)800 58875 y(for)411 b(all)h(ro)-36 b(ws)411 b(un)-36 b(til)410 b(the)g(\()p Fr(n)249 b Fq(\241)g Ft(1\)-th)410 b(one.)570 b(By)412 b(Lemma)p 0 .5 0 TeXcolorrgb 28358 58875 a SDict begin H.S end 28358 58875 a 0 .5 0 TeXcolorrgb Ft(7)p 0 .5 0 TeXcolorrgb 29008 58019 a SDict begin H.R end 29008 58019 a 29008 58875 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.7) cvn H.B /ANN pdfmark end 29008 58875 a Black Ft(,)k(w)-36 b(e)411 b(ha)-36 b(v)g(e)411 b(Card\()p Fr(\275)38648 59074 y Fy(n)p Fw(\241)p Fv(1)40476 58875 y Ft(\))369 b(=)g(Card\()p Fr(c)46616 58393 y Fv(+)47403 58875 y Ft(\()p Fr(\275)48580 59074 y Fy(n)p Fw(\241)p Fv(1)50408 58875 y Ft(\)\))f(=)800 60480 y(Card\()p Fr(\275)4795 59998 y Fv(+)4795 60809 y Fy(n)5582 60480 y Ft(\),)538 b(so)518 b(Card\()p Fr(\275)12663 59998 y Fv(+)12663 60809 y Fy(n)13450 60480 y Ft(\))511 b(=)g Fr(F)16832 60679 y Fy(n)p Fw(\241)p Fv(1)18661 60480 y Ft(.)829 b(By)518 b(Lemmas)p 0 .5 0 TeXcolorrgb 27215 60480 a SDict begin H.S end 27215 60480 a 0 .5 0 TeXcolorrgb Ft(8)p 0 .5 0 TeXcolorrgb 27865 59624 a SDict begin H.R end 27865 59624 a 27865 60480 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.8) cvn H.B /ANN pdfmark end 27865 60480 a Black 517 w Ft(and)p 0 .5 0 TeXcolorrgb 30995 60480 a SDict begin H.S end 30995 60480 a 0 .5 0 TeXcolorrgb Ft(9)p 0 .5 0 TeXcolorrgb 31645 59624 a SDict begin H.R end 31645 59624 a 31645 60480 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.9) cvn H.B /ANN pdfmark end 31645 60480 a Black Ft(,)539 b(w)-36 b(e)517 b(ha)-36 b(v)g(e)518 b(Card\()p Fr(\275)41621 59998 y Fw(\241)41621 60809 y Fy(n)42408 60480 y Ft(\))511 b(=)g(Card\()p Fr(\275)48943 59931 y Fv(+)48943 60806 y Fy(n)p Fw(\241)p Fv(1)50771 60480 y Ft(\))g(=)800 62086 y(Card\()p 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Fy(n)14350 66403 y Ft(=)368 b(2)p Fr(G)17406 66602 y Fy(n)p Fw(\241)p Fv(1)19530 66403 y Ft(+)295 b Fr(G)21863 66602 y Fy(n)p Fw(\241)p Fv(3)23692 66403 y Fs(.)p Black 26475 70521 a Ft(9)p Black eop %%Page: 10 10 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 1328 x Fs(Pr)-66 b(o)g(of.)p Black 649 w Ft(W)-108 b(e)552 b(write)h Fr(G)11699 -3154 y Fw(\241)11699 -2343 y Fy(n)13039 -2672 y Ft(\(resp.)933 b Fr(G)18185 -3154 y Fv(+)18185 -2343 y Fy(n)18973 -2672 y Ft(\))552 b(for)h Fr(S)77 b Ft(\()p Fr(\275)24191 -3154 y Fw(\241)24191 -2343 y Fy(n)24978 -2672 y Ft(\))552 b(\(resp.)934 b Fr(S)77 b Ft(\()p Fr(\275)32211 -3154 y Fv(+)32211 -2343 y Fy(n)32998 -2672 y Ft(\)\).)934 b(Lemmas)p 0 .5 0 TeXcolorrgb 40579 -2672 a SDict begin H.S end 40579 -2672 a 0 .5 0 TeXcolorrgb Ft(5)p 0 .5 0 TeXcolorrgb 41229 -3528 a SDict begin H.R end 41229 -3528 a 41229 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.5) cvn H.B /ANN pdfmark end 41229 -2672 a Black Ft(,)p 0 .5 0 TeXcolorrgb 42173 -2672 a SDict begin H.S end 42173 -2672 a 0 .5 0 TeXcolorrgb Ft(6)p 0 .5 0 TeXcolorrgb 42823 -3528 a SDict begin H.R end 42823 -3528 a 42823 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.6) cvn H.B /ANN pdfmark end 42823 -2672 a Black 552 w Ft(and)p 0 .5 0 TeXcolorrgb 46023 -2672 a SDict begin H.S end 46023 -2672 a 0 .5 0 TeXcolorrgb Ft(9)p 0 .5 0 TeXcolorrgb 46673 -3528 a SDict begin H.R end 46673 -3528 a 46673 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.9) cvn H.B /ANN pdfmark end 46673 -2672 a Black 553 w Ft(imply)553 b(the)800 -1067 y(relation)508 b Fr(G)6669 -1549 y Fw(\241)6669 -738 y Fy(n)7950 -1067 y Ft(=)494 b Fr(G)10482 -1616 y Fv(+)10482 -741 y Fy(n)p Fw(\241)p Fv(1)12656 -1067 y Fq(\241)345 b Fr(G)15060 -868 y Fy(n)p Fw(\241)p Fv(2)16889 -1067 y Ft(.)799 b(W)-108 b(e)507 b(split)g(the)g(edges)g(of)h Fr(\275)31422 -1549 y Fv(+)31422 -738 y Fy(n)32716 -1067 y Ft(in)f(t)-36 b(w)g(o)508 b(subsets:)724 b(the)507 b(one,)526 b(sa)-36 b(y)508 b Fr(\276)50252 -1549 y Fv(+)50204 -738 y Fy(n)51039 -1067 y Ft(,)526 b(is)800 539 y(comp)36 b(osed)374 b(b)-36 b(y)374 b(the)f(edges)h(whic)-36 b(h)373 b(are)h(c)-36 b(hildren)373 b(of)i(elemen)-36 b(ts)373 b(of)i Fr(\275)33484 -11 y Fv(+)33484 864 y Fy(n)p Fw(\241)p Fv(1)35312 539 y Ft(,)386 b(the)373 b(other,)386 b Fr(\276)42736 56 y Fw(\241)42688 867 y Fy(n)43523 539 y Ft(,)g(is)374 b(comp)36 b(osed)374 b(b)-36 b(y)800 2144 y(the)304 b(edges)h(whic)-36 b(h)304 b(are)h(c)-36 b(hildren)303 b(of)j(elemen)-36 b(ts)304 b(of)h Fr(\275)25274 1594 y Fw(\241)25274 2469 y Fy(n)p Fw(\241)p Fv(1)27103 2144 y Ft(.)535 b(Lemma)p 0 .5 0 TeXcolorrgb 32512 2144 a SDict begin H.S end 32512 2144 a 0 .5 0 TeXcolorrgb Ft(7)p 0 .5 0 TeXcolorrgb 33163 1288 a SDict begin H.R end 33163 1288 a 33163 2144 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.7) cvn H.B /ANN pdfmark end 33163 2144 a Black 304 w Ft(implies)305 b(that)f Fr(S)77 b Ft(\()p Fr(\276)42612 1662 y Fv(+)42564 2472 y Fy(n)43399 2144 y Ft(\))369 b(=)g Fr(G)46681 1594 y Fv(+)46681 2469 y Fy(n)p Fw(\241)p Fv(1)48541 2144 y Ft(+)32 b Fr(G)50611 2343 y Fy(n)p Fw(\241)p Fv(2)52439 2144 y Ft(,)800 3749 y(Lemmas)p 0 .5 0 TeXcolorrgb 5956 3749 a SDict begin H.S end 5956 3749 a 0 .5 0 TeXcolorrgb Ft(5)p 0 .5 0 TeXcolorrgb 6606 2893 a SDict begin H.R end 6606 2893 a 6606 3749 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.5) cvn H.B /ANN pdfmark end 6606 3749 a Black 433 w Ft(and)p 0 .5 0 TeXcolorrgb 9568 3749 a SDict begin H.S end 9568 3749 a 0 .5 0 TeXcolorrgb Ft(9)p 0 .5 0 TeXcolorrgb 10219 2893 a SDict begin H.R end 10219 2893 a 10219 3749 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.9) cvn H.B /ANN pdfmark end 10219 3749 a Black 433 w Ft(that)433 b Fr(S)77 b Ft(\()p Fr(\276)15640 3267 y Fw(\241)15592 4077 y Fy(n)16428 3749 y Ft(\))368 b(=)h Fr(G)19709 3199 y Fw(\241)19709 4074 y Fy(n)p Fw(\241)p Fv(1)21833 3749 y Ft(+)295 b Fr(G)24166 3199 y Fv(+)24166 4074 y Fy(n)p Fw(\241)p Fv(2)25994 3749 y Ft(.)578 b(Th)-36 b(us,)434 b(w)-36 b(e)434 b(get)6775 6494 y Fr(G)7801 6693 y Fy(n)8797 6494 y Ft(=)368 b Fr(G)11203 5946 y Fw(\241)11203 6823 y Fy(n)12286 6494 y Ft(+)294 b Fr(G)14618 5946 y Fv(+)14618 6823 y Fy(n)16513 6494 y Ft(=)1106 b Fr(G)19657 5946 y Fw(\241)19657 6823 y Fy(n)20739 6494 y Ft(+)295 b Fr(S)77 b Ft(\()p Fr(\276)24216 5946 y Fw(\241)24168 6823 y Fy(n)25003 6494 y Ft(\))295 b(+)g Fr(S)77 b Ft(\()p Fr(\276)29281 5946 y Fv(+)29233 6823 y Fy(n)30068 6494 y Ft(\))16513 8432 y(=)1106 b(\()p Fr(G)20163 7883 y Fv(+)20163 8760 y Fy(n)p Fw(\241)p Fv(1)22287 8432 y Fq(\241)295 b Fr(G)24641 8631 y Fy(n)p Fw(\241)p Fv(2)26469 8432 y Ft(\))g(+)g(\()p Fr(G)30109 7883 y Fw(\241)30109 8760 y Fy(n)p Fw(\241)p Fv(1)32233 8432 y Ft(+)g Fr(G)34566 7883 y Fv(+)34566 8760 y Fy(n)p Fw(\241)p Fv(2)36394 8432 y Ft(\))g(+)g(\()p Fr(G)40034 7883 y Fv(+)40034 8760 y Fy(n)p Fw(\241)p Fv(1)42157 8432 y Ft(+)g Fr(G)44490 8631 y Fy(n)p Fw(\241)p Fv(2)46319 8432 y Ft(\))16513 10369 y(=)1106 b(2)p Fr(G)20307 9820 y Fv(+)20307 10697 y Fy(n)p Fw(\241)p Fv(1)22431 10369 y Ft(+)295 b Fr(G)24764 9820 y Fw(\241)24764 10697 y Fy(n)p Fw(\241)p Fv(1)26888 10369 y Ft(+)f Fr(G)29220 9820 y Fv(+)29220 10697 y Fy(n)p Fw(\241)p Fv(2)16513 12306 y Ft(=)1106 b(2)p Fr(G)20307 12505 y Fy(n)p Fw(\241)p Fv(1)22431 12306 y Ft(+)295 b Fr(G)24764 11757 y Fv(+)24764 12634 y Fy(n)p Fw(\241)p Fv(2)26888 12306 y Fq(\241)g Fr(G)29242 11757 y Fw(\241)29242 12634 y Fy(n)p Fw(\241)p Fv(1)16513 14243 y Ft(=)1106 b(2)p Fr(G)20307 14442 y Fy(n)p Fw(\241)p Fv(1)22431 14243 y Ft(+)295 b Fr(G)24764 14442 y Fy(n)p Fw(\241)p Fv(3)26592 14243 y Fr(;)800 16989 y Ft(the)433 b(relation)h Fr(G)8835 16439 y Fv(+)8835 17314 y Fy(n)p Fw(\241)p Fv(2)10959 16989 y Fq(\241)295 b Fr(G)13313 16439 y Fw(\241)13313 17314 y Fy(n)p Fw(\241)p Fv(1)15511 16989 y Ft(=)368 b Fr(G)17917 17188 y Fy(n)p Fw(\241)p Fv(3)20179 16989 y Ft(coming)434 b(from)g(Lemmas)p 0 .5 0 TeXcolorrgb 32886 16989 a SDict begin H.S end 32886 16989 a 0 .5 0 TeXcolorrgb Ft(5)p 0 .5 0 TeXcolorrgb 33536 16133 a SDict begin H.R end 33536 16133 a 33536 16989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.5) cvn H.B /ANN pdfmark end 33536 16989 a Black Ft(,)p 0 .5 0 TeXcolorrgb 34331 16989 a SDict begin H.S end 34331 16989 a 0 .5 0 TeXcolorrgb Ft(6)p 0 .5 0 TeXcolorrgb 34981 16133 a SDict begin H.R end 34981 16133 a 34981 16989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.6) cvn H.B /ANN pdfmark end 34981 16989 a Black Ft(,)p 0 .5 0 TeXcolorrgb 35776 16989 a SDict begin H.S end 35776 16989 a 0 .5 0 TeXcolorrgb Ft(7)p 0 .5 0 TeXcolorrgb 36427 16133 a SDict begin H.R end 36427 16133 a 36427 16989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.7) cvn H.B /ANN pdfmark end 36427 16989 a Black 433 w Ft(and)p 0 .5 0 TeXcolorrgb 39389 16989 a SDict begin H.S end 39389 16989 a 0 .5 0 TeXcolorrgb Ft(9)p 0 .5 0 TeXcolorrgb 40040 16133 a SDict begin H.R end 40040 16133 a 40040 16989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.9) cvn H.B /ANN pdfmark end 40040 16989 a Black Ft(.)p 51860 16989 45 878 v 51905 16155 781 45 v 51905 16989 V 52684 16989 45 878 v 2751 19536 a(The)478 b(same)h(kind)f(of)h(study)f(sho)-36 b(ws)479 b(that,)489 b(more)479 b(generally)-108 b(,)490 b(if)479 b(w)-36 b(e)479 b(write)g Fr(v)40243 19735 y Fy(n)41314 19536 y Ft(=)445 b Fr(\256)43598 19735 y Fy(n)44224 19536 y Fr(a)326 b Ft(+)f Fr(\257)47304 19735 y Fy(n)47930 19536 y Fr(b)479 b Ft(for)g(the)800 21141 y(sum)433 b(of)h(the)f Fr(n)p Ft(-th)g(ro)-36 b(w)434 b(of)h(the)e(general)h(restricted)f (tree,)g(then)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)3769 23887 y Fr(\256)4596 24086 y Fv(0)5491 23887 y Ft(=)369 b(1)1301 b Fr(\256)9650 24086 y Fv(1)10545 23887 y Ft(=)369 b(0)1301 b Fr(\256)14704 24086 y Fv(2)15599 23887 y Ft(=)368 b(1)1301 b Fr(\256)19757 24086 y Fv(3)20652 23887 y Ft(=)369 b(2)1301 b Fr(\256)24811 24086 y Fy(n)25807 23887 y Ft(=)368 b(2)p Fr(\256)28664 24086 y Fy(n)p Fw(\241)p Fv(1)30789 23887 y Ft(+)294 b Fr(\256)32922 24086 y Fy(n)p Fw(\241)p Fv(3)35185 23887 y Ft(for)434 b Fr(n)369 b Fq(\270)g Ft(4)434 b(\(so)g Fr(\256)43733 24086 y Fy(n)44729 23887 y Ft(=)368 b Fr(G)47135 24086 y Fy(n)p Fw(\241)p Fv(1)48964 23887 y Ft(\),)8645 26633 y Fr(\257)9379 26832 y Fv(0)10274 26633 y Ft(=)g(0)1301 b Fr(\257)14339 26832 y Fv(1)15234 26633 y Ft(=)369 b(1)1301 b Fr(\257)19300 26832 y Fv(2)20195 26633 y Ft(=)369 b(1)1301 b Fr(\257)24261 26832 y Fv(3)25156 26633 y Ft(=)368 b(2)1301 b Fr(\257)29221 26832 y Fy(n)30217 26633 y Ft(=)369 b(2)p Fr(\257)32982 26832 y Fy(n)p Fw(\241)p Fv(1)35106 26633 y Ft(+)295 b Fr(\257)37147 26832 y Fy(n)p Fw(\241)p Fv(3)39409 26633 y Ft(for)434 b Fr(n)370 b Fq(\270)f Ft(3)p Fr(:)800 27563 y SDict begin H.S end 800 27563 a 800 27563 a SDict begin 14.5 H.A end 800 27563 a 800 27563 a SDict begin [ /View [/XYZ H.V] /Dest (Cor.3) cvn H.B /DEST pdfmark end 800 27563 a Black 1621 x Fo(Corollary)607 b(3.)p Black 592 w Fs(L)-66 b(et)533 b(us)i(denote)e(by)778 b Ft(~)-894 b Fr(m)20495 29383 y Fy(n)21655 29184 y Fs(the)534 b(aver)-66 b(age)533 b(value)h(of)g(an)g(element)e(of)i(the)g Fr(n)p Fs(-th)g(r)-66 b(ow)534 b(of)g Fo(R)p Fs(.)800 30789 y(As)475 b Fr(n)g Fs(go)-66 b(es)474 b(to)h(in\257nity,)f(the)h(r)-66 b(atio)717 b Ft(~)-893 b Fr(m)19742 30988 y Fy(n)p Fv(+1)21570 30789 y Fr(=)244 b Ft(~)-894 b Fr(m)23358 30988 y Fy(n)24459 30789 y Fs(tends)474 b(to)h Fr(\256)8 b(=')388 b Fq(\274)g Ft(1)p Fr(:)p Ft(363117)p Fs(,)479 b(wher)-66 b(e)475 b Fr(\256)483 b Fs(is)475 b(the)f(only)h(r)-66 b(e)g(al)800 32394 y(zer)g(o)465 b(of)f(the)h(e)-66 b(quation)464 b Fr(x)13146 31912 y Fv(3)14041 32394 y Ft(=)368 b(2)p Fr(x)16810 31912 y Fv(2)17632 32394 y Ft(+)294 b(1)466 b Fs(and)f Fr(')g Fs(the)f(golden)h(r)-66 b(atio.)p Black 800 34945 a(Pr)g(o)g(of.)p Black 649 w Ft(Classical)496 b(facts)f(ab)36 b(out)494 b(linear)g(recurring)g(sequences)g(sho)-36 b(w)494 b(that)g(the)f(ratio)i Fr(G)45133 35144 y Fy(n)p Fv(+1)46961 34945 y Fr(=G)48637 35144 y Fy(n)49758 34945 y Ft(tends)800 36550 y(to)394 b Fr(\256)402 b Ft(as)394 b Fr(n)g Ft(tends)f(to)h(in\257nit)-36 b(y)-108 b(.)564 b(Since)393 b(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 26471 36550 a SDict begin H.S end 26471 36550 a 0 .5 0 TeXcolorrgb Ft(4)p 0 .5 0 TeXcolorrgb 27121 35694 a SDict begin H.R end 27121 35694 a 27121 36550 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.4) cvn H.B /ANN pdfmark end 27121 36550 a Black 394 w Ft(sho)-36 b(ws)394 b(that)636 b(~)-893 b Fr(m)35127 36749 y Fy(n)36122 36550 y Ft(=)368 b Fr(G)38528 36749 y Fy(n)39155 36550 y Fr(=F)40647 36749 y Fy(n)41273 36550 y Ft(,)402 b(w)-36 b(e)394 b(get)638 b(~)-894 b Fr(m)47177 36749 y Fy(n)p Fv(+1)49005 36550 y Fr(=)244 b Ft(~)-894 b Fr(m)50793 36749 y Fy(n)51788 36550 y Ft(=)800 38155 y(\()p Fr(G)2332 38354 y Fy(n)p Fv(+1)4160 38155 y Fr(=G)5836 38354 y Fy(n)6463 38155 y Ft(\))354 b Fq(\242)h Ft(\()p Fr(F)9395 38354 y Fy(n)p Fv(+1)11223 38155 y Fr(=F)12715 38354 y Fy(n)13341 38155 y Ft(\))521 b(and)f(the)g(result)g(follo)-36 b(ws)523 b(from)e(the)f(fact)h(that)f Fr(F)39260 38354 y Fy(n)p Fv(+1)41089 38155 y Fr(=F)42581 38354 y Fy(n)43728 38155 y Ft(tends)f(to)i Fr(')g Ft(as)g Fr(n)800 39760 y Ft(go)36 b(es)435 b(to)e(in\257nit)-36 b(y)-108 b(.)p 51860 39760 45 878 v 51905 38927 781 45 v 51905 39760 V 52684 39760 45 878 v 800 41850 a SDict begin H.S end 800 41850 a 800 41850 a SDict begin 14.5 H.A end 800 41850 a 800 41850 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.4) cvn H.B /DEST pdfmark end 800 41850 a 1734 x Fn(2.4)1793 b(Pro)100 b(jection)598 b(of)f(w)-50 b(alks)599 b(of)e(T)h(in)-50 b(to)599 b(w)-50 b(alks)598 b(of)g(R)800 46047 y Ft(A)420 b(\(\257nite\))f(w)-36 b(alk)422 b(in)e Fo(T)g Ft(can)g(b)36 b(e)420 b(represen)-36 b(ted)419 b(as)i(a)f(\257nite)g(sequence)g(\()p Fr(w)36303 46246 y Fy(i)36678 46047 y Ft(\))37184 46246 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)40755 46047 y Ft(of)h(+)f(and)f Fq(\241)i Ft(signs.)574 b(T)-108 b(o)800 47652 y(suc)-36 b(h)470 b(a)h(sequence)f(is)h(asso)36 b(ciated)471 b(the)f(random)g(Fib)36 b(onacci)471 b(sequence)g(\()p Fr(g)37513 47851 y Fy(n)38139 47652 y Ft(\))38645 47851 y Fy(n)39741 47652 y Ft(de\257ned)e(b)-36 b(y)471 b Fr(g)46725 47851 y Fv(0)47682 47652 y Ft(=)432 b Fr(g)49749 47851 y Fv(1)50706 47652 y Ft(=)g(1)800 49257 y(and,)477 b(for)470 b(all)f Fr(n)430 b Fq(\270)f Ft(2,)478 b Fr(g)12378 49456 y Fy(n)13433 49257 y Ft(=)428 b Fr(g)15496 49456 y Fy(n)p Fw(\241)p Fv(1)17644 49257 y Ft(+)319 b Fr(g)19598 49456 y Fy(n)p Fw(\241)p Fv(2)21895 49257 y Ft(i\256)469 b Fr(w)24414 49456 y Fy(n)p Fw(\241)p Fv(2)26671 49257 y Ft(=)428 b(+.)684 b(A)469 b(\257nite)f(w)-36 b(alk)470 b(\()p Fr(w)39477 49456 y Fy(i)39852 49257 y Ft(\))40358 49456 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)43978 49257 y Ft(b)36 b(eing)468 b(giv)-36 b(en,)479 b(w)-36 b(e)800 50862 y(generically)371 b(denote)d(b)-36 b(y)369 b(\()p Fr(g)14200 51061 y Fy(i)14575 50862 y Ft(\))15081 51061 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)18601 50862 y Ft(the)g(asso)36 b(ciated)370 b(random)e(Fib)36 b(onacci)370 b(sequence,)382 b(that)369 b(is)g(the)g(v)-72 b(alues)800 52467 y(of)434 b(the)f(no)36 b(des)434 b(successiv)-36 b(ely)435 b(attained)e(in)g Fo(T)h Ft(b)-36 b(y)434 b(this)f(w)-36 b(alk.)2751 54072 y(The)434 b(follo)-36 b(wing)435 b(prop)36 b(osition)434 b(giv)-36 b(es)435 b(a)f(w)-36 b(a)g(y)434 b(to)g(transform)f(a)h(w)-36 b(alk)435 b(in)f Fo(T)f Ft(in)-36 b(to)434 b(a)g(w)-36 b(alk)435 b(on)e Fo(R)p Ft(.)800 54331 y SDict begin H.S end 800 54331 a 800 54331 a SDict begin 14.5 H.A end 800 54331 a 800 54331 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.10) cvn H.B /DEST pdfmark end 800 54331 a Black 2292 x Fo(Lemma)441 b(10.)p Black 516 w Fs(F)-100 b(or)419 b Fr(N)508 b Fq(\270)369 b Ft(2)p Fs(,)428 b(let)418 b Ft(\()p Fr(w)18486 56822 y Fy(n)19112 56623 y Ft(\))19618 56822 y Fv(0)p Fw(\267)p Fy(n)p Fw(\267)p Fy(N)23438 56623 y Fs(b)-66 b(e)418 b(a)h(\257nite)e(walk)j(in)e Fo(T)h Fs(such)g(that)g(a)g(minus)f(sign)g (is)h(never)800 58229 y(fol)66 b(lowe)-66 b(d)557 b(by)e(another)h (minus)g(sign,)577 b(ap)-66 b(art)556 b(for)g Fr(w)26285 58428 y Fy(N)94 b Fw(\241)p Fv(1)28940 58229 y Fs(and)556 b Fr(w)32485 58428 y Fy(N)33937 58229 y Fs(\(b)-66 b(oth)556 b(minus)f(signs\).)871 b(Denoting)554 b(by)800 59834 y Ft(\()p Fr(g)1929 60033 y Fy(i)2304 59834 y Ft(\))2810 60033 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)94 b Fv(+2)7628 59834 y Fs(the)465 b(c)-66 b(orr)g(esp)g(onding)463 b(no)-66 b(de)464 b(se)-66 b(quenc)g(e,)463 b(we)i(have)g Ft(\()p Fr(g)32768 60033 y Fy(N)94 b Fv(+2)34866 59834 y Fr(;)221 b(g)36071 60033 y Fy(N)94 b Fv(+1)38170 59834 y Ft(\))369 b(=)g(\()p Fr(g)41555 60033 y Fy(N)94 b Fw(\241)p Fv(1)43653 59834 y Fr(;)221 b(g)44858 60033 y Fy(N)45755 59834 y Ft(\))p Fs(.)p Black 800 62385 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(The)373 b(assumption)f(made)h(on)g(\()p Fr(w)20893 62584 y Fy(i)21268 62385 y Ft(\))21774 62584 y Fv(0)p Fw(\267)p Fy(i)e Ft(3)p Fs(,)465 b(we)g(have)g Fr(s)p Ft(\()p Fr(\275)23588 30598 y Fy(n)24213 30399 y Ft(\))369 b(=)g Fr(\275)27140 30598 y Fy(n)p Fv(+1)29263 30399 y Ft(+)295 b Fr(\275)31241 30598 y Fy(n)p Fw(\241)p Fv(2)33070 30399 y Fs(.)2751 33368 y Ft(Let)383 b(us)g(consider)g(no)-36 b(w)384 b(the)e(b)36 b(eginning)384 b(of)g(the)f(trees)g Fo(T)30150 33575 y Fv(\(1)p Fy(;')p Fv(\))32666 33368 y Ft(and)f Fo(T)36183 33591 y Fv(\(1)p Fy(;')37893 33339 y Fg(\241)p Fl(1)38995 33591 y Fv(\))39800 33368 y Ft(\(where)h Fr(\275)44684 33567 y Fv(2)45579 33368 y Ft(=)368 b Fr(\275)47630 32689 y Fv(\()p Fy(')48609 32377 y Fg(\241)p Fl(1)49710 32689 y Fy(;)p Fv(1\))47630 33694 y(2)51246 33368 y Ft(for)800 35342 y(the)410 b(left)i(\257gure)e(and)g Fr(\275)12038 35541 y Fv(2)12933 35342 y Ft(=)369 b Fr(\275)14985 34663 y Fv(\()p Fy(')15964 34351 y Fg(\241)p Fl(2)17064 34663 y Fy(;)p Fv(1\))14985 35668 y(2)18628 35342 y Ft(for)412 b(the)e(righ)-36 b(t)411 b(one,)k(with)c(the)g(previous)g(abuse)f(of)i (notation)f(in)f(this)800 36947 y(latter)434 b(case\).)p Black 10689 45239 a @beginspecial @setspecial 5 5 sub 1864679 mul 65536 div 72.27 div 72 mul 7 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 x @beginspecial @setspecial 3 5 sub 1864679 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72.27 div 72 mul [ ] true psrectangle @endspecial 12598 w @beginspecial @setspecial 0.410 1864679 mul 65536 div 72.27 div 72 mul 0.403 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 32736 57838 a @beginspecial @setspecial 0.809 1864679 mul 65536 div 72.27 div 72 mul 0.556 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6300 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6299 w @beginspecial @setspecial 0.346 1864679 mul 65536 div 72.27 div 72 mul 0.411 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6299 w @beginspecial @setspecial 1.588 1864679 mul 65536 div 72.27 div 72 mul 0.556 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 10364 45667 a(1)10263 48546 y Fr(')3335 51936 y(')4187 51454 y Fw(\241)p Fv(1)16299 51936 y Fr(')17151 51454 y Fv(2)915 58266 y Ft(1)5258 58235 y Fr(')295 b Ft(+)g Fr(')8564 57753 y Fw(\241)p Fv(1)13513 58266 y Ft(1)19449 58235 y Fr(')20301 57753 y Fv(3)10689 46814 y @beginspecial @setspecial 894951 65536 div 677998 65536 div psdeletebox @endspecial 10091 46971 a Fr(\275)10762 47170 y Fv(2)2815 54688 y @beginspecial @setspecial 2143961 65536 div 962660 65536 div psdeletebox @endspecial 1162 55085 a Fr(')2014 54603 y Fw(\241)p Fv(1)3272 55085 y Fr(\275)3943 55284 y Fv(2)41860 45667 y Ft(1)41130 48786 y Fr(')41982 48304 y Fw(\241)p Fv(1)34831 51936 y Fr(')35683 51454 y Fw(\241)p Fv(2)48058 51695 y Fr(')31681 58235 y(')32533 57753 y Fw(\241)p Fv(3)38710 58266 y Ft(1)5650 b(1)49352 58235 y Fr(')295 b Ft(+)g Fr(')52658 57753 y Fw(\241)p Fv(1)42185 46814 y @beginspecial @setspecial 894951 65536 div 677998 65536 div psdeletebox @endspecial 41587 46971 a Fr(\275)42258 47170 y Fv(2)34311 54688 y @beginspecial @setspecial 2143961 65536 div 962660 65536 div psdeletebox @endspecial 32658 55085 a Fr(')33510 54603 y Fw(\241)p Fv(2)34768 55085 y Fr(\275)35439 55284 y Fv(2)800 60987 y SDict begin H.S end 800 60987 a 800 60987 a SDict begin 14.5 H.A end 800 60987 a 800 60987 a SDict begin [ /View [/XYZ H.V] /Dest (figure.3) cvn H.B /DEST pdfmark end 800 60987 a Black 15677 63699 a Ft(Figure)434 b(3:)p 0 TeXcolorgray 21367 62094 a SDict begin H.S end 21367 62094 a 0 TeXcolorgray 0 TeXcolorgray 21367 62094 a SDict begin H.R end 21367 62094 a 21367 62094 a SDict begin [ /View [/XYZ H.V] /Dest (figure.3) cvn H.B /DEST pdfmark end 21367 62094 a Black 1605 x Ft(The)g(trees)f Fo(T)28194 63906 y Fv(\(1)p Fy(;')p Fv(\))30760 63699 y Ft(and)g Fo(T)34328 63922 y Fv(\(1)p Fy(;')36038 63670 y Fg(\241)p Fl(1)37140 63922 y Fv(\))37562 63699 y Ft(.)p Black Black Black 26150 70521 a(12)p Black eop %%Page: 13 13 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 2751 -2657 a Ft(W)-108 b(e're)341 b(in)-36 b(terested)339 b(in)h(the)g(sequence)h Fr(s)21700 -3139 y Fy(n)22326 -2657 y Ft(\()p Fr(\275)23503 -3336 y Fv(\()p Fw(j)p Fy(b)p Fw(\241)p Fy(a)p Fw(j)p Fy(;a)p Fv(\))23503 -2331 y(2)27208 -2657 y Ft(\))f(for)h(\()p Fr(a;)221 b(b)p Ft(\))369 b(=)f(\(1)p Fr(;)221 b(')p Ft(\))342 b(and)d(\(1)p Fr(;)221 b(')42990 -3139 y Fw(\241)p Fv(1)44249 -2657 y Ft(\).)547 b(Let)340 b(us)g(de\257ne)800 -1052 y Fr(z)518 b Ft(as)459 b Fr(z)470 b Ft(:=)412 b Fq(j)p Fr(b)312 b Fq(\241)g Fr(a)p Fq(j)p Ft(:)628 b(w)-36 b(e)459 b(th)-36 b(us)457 b(ha)-36 b(v)g(e)459 b Fr(z)470 b Ft(=)411 b Fr(')22222 -1534 y Fw(\241)p Fv(1)23938 -1052 y Ft(for)459 b Fr(b)412 b Ft(=)f Fr(')458 b Ft(and)g Fr(z)471 b Ft(=)411 b Fr(')35552 -1534 y Fw(\241)p Fv(2)37268 -1052 y Ft(for)459 b Fr(b)411 b Ft(=)g Fr(')42519 -1534 y Fw(\241)p Fv(1)43777 -1052 y Ft(.)653 b(T)-108 b(ogether)458 b(with)800 553 y(the)433 b(rules)g(giv)-36 b(en)435 b(in)e(Lemma)p 0 .5 0 TeXcolorrgb 15710 553 a SDict begin H.S end 15710 553 a 0 .5 0 TeXcolorrgb Ft(11)p 0 .5 0 TeXcolorrgb 17011 -303 a SDict begin H.R end 17011 -303 a 17011 553 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.11) cvn H.B /ANN pdfmark end 17011 553 a Black Ft(,)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing)435 b(rules)f(for)g Fr(b)369 b Ft(=)f Fr(')434 b Ft(or)g Fr(')41583 71 y 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b(Dividing)431 b(b)-36 b(y)430 b(2)g(the)f(v)-72 b(alue)800 11197 y(2\()p Fr(\256)202 b Fq(\241)194 b Ft(1\))385 b(to)f(get)g(the)f(a)-36 b(v)g(erage)385 b(v)-72 b(alue)385 b(of)g(the)e Fr(n)p Ft(-th)g(term,)394 b(w)-36 b(e)384 b(obtain)g(the)f(result)h(stated)g (in)f(Theorem)p 0 .5 0 TeXcolorrgb 51788 11197 a SDict begin H.S end 51788 11197 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 52439 10341 a SDict begin H.R end 52439 10341 a 52439 11197 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 52439 11197 a Black Ft(.)800 13518 y SDict begin H.S end 800 13518 a 800 13518 a SDict begin 14.5 H.A end 800 13518 a 800 13518 a SDict begin [ /View [/XYZ H.V] /Dest (section.4) cvn H.B /DEST pdfmark end 800 13518 a 2103 x Fu(4)2152 b(In)-60 b(teresting)716 b(prop)60 b(erties)715 b(of)i(R)g(and)g(related)f (trees)800 18542 y Ft(This)515 b(section)h(is)f(dev)-36 b(oted)515 b(to)g(some)h(prop)36 b(erties)514 b(of)i Fo(R)f Ft(whic)-36 b(h)515 b(seem)g(of)h(in)-36 b(terest)514 b(to)i(us.)822 b(It)515 b(concerns)800 20147 y(the)579 b(arithmetical)h(prop)36 b(erties)579 b(of)h(the)f(no)36 b(des)579 b(of)h(the)f(tree,)616 b(some)579 b(facts)h(ab)36 b(out)579 b(con)-36 b(tin)g(ued)578 b(fraction)800 21752 y(expansion,)410 b(some)403 b(com)-36 b(binatorial)403 b(prop)36 b(erties,)409 b(and)402 b(also,)411 b(for)403 b(an)g(imp)36 b(ortan)-36 b(t)402 b(part,)409 b(the)402 b(link)i(b)36 b(et)-36 b(w)g(een)800 23357 y(the)433 b(tree)g(and)g(the)h(sets)f(SL)o(\(2)p Fr(;)221 b Fd(N)p Ft(\))435 b(and)e(SL)o(\(2)p Fr(;)221 b Fd(Z)p Ft(\).)800 25536 y SDict begin H.S end 800 25536 a 800 25536 a SDict begin 14.5 H.A end 800 25536 a 800 25536 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.1) cvn H.B /DEST pdfmark end 800 25536 a 1660 x Fn(4.1)1793 b(V)-149 b(alues)598 b(in)g(the)h Fk(n)p Fn(-th)f(ro)-50 b(w)598 b(of)f(R)800 29659 y Ft(Lemma)p 0 .5 0 TeXcolorrgb 5415 29659 a SDict begin H.S end 5415 29659 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 6066 28803 a SDict begin H.R end 6066 28803 a 6066 29659 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.1) cvn H.B /ANN pdfmark end 6066 29659 a Black 406 w Ft(allo)-36 b(ws)408 b(to)f(kno)-36 b(w)407 b(explicitly)h(the)d (n)-36 b(um)g(b)36 b(ers)405 b(of)i(1s)g(in)g(an)-36 b(y)406 b(ro)-36 b(w.)570 b(Our)405 b(aim)i(is)g(here)f(to)g(extend)800 31264 y(this)433 b(result)h(to)f(the)g(other)g(n)-36 b(um)g(b)36 b(ers.)2751 32869 y(Let)606 b(us)f(recall)i(that)f(the)f Fs(Euler)624 b(function)605 b Fr(\301)g Ft(maps)h(eac)-36 b(h)606 b(p)36 b(ositiv)-36 b(e)607 b(in)-36 b(teger)606 b Fr(n)h Ft(to)f(the)f(n)-36 b(um)g(b)36 b(er,)800 34474 y(denoted)447 b(b)-36 b(y)448 b Fr(\301)p Ft(\()p Fr(n)p Ft(\),)j(of)e(p)36 b(ositiv)-36 b(e)449 b(in)-36 b(tegers)447 b Fr(k)439 b(<)393 b(n)448 b Ft(suc)-36 b(h)447 b(that)g Fr(k)493 b Ft(and)448 b Fr(n)g Ft(are)g(relativ)-36 b(ely)449 b(prime.)621 b(\(W)-108 b(e)448 b(\257x)800 36079 y(the)433 b(con)-36 b(v)g(en)g(tion)433 b Fr(\301)p Ft(\(1\))369 b(=)g(1.\))800 36411 y SDict begin H.S end 800 36411 a 800 36411 a SDict begin 14.5 H.A end 800 36411 a 800 36411 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.13) cvn H.B /DEST pdfmark end 800 36411 a Black 2308 x Fo(Lemma)498 b(13.)p Black 553 w Fs(L)-66 b(et)463 b Fr(n)369 b Fq(\270)g Ft(1)464 b Fs(b)-66 b(e)463 b(an)h(inte)-66 b(ger.)595 b(The)463 b(no)-66 b(des)464 b(in)f Fo(R)g Fs(with)h(value)g Fr(n)g Fs(ar)-66 b(e)464 b(include)-66 b(d)463 b(in)g(a)h(union)800 40324 y(of)h Ft(2)p Fr(\301)p Ft(\()p Fr(n)p Ft(\))g Fs(walks)g(in)g Fo(R)p Fs(.)p Black 800 42964 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(The)516 b(case)g Fr(n)508 b Ft(=)h(1)515 b(is)h(con)-36 b(tained)515 b(in)h(Lemma)p 0 .5 0 TeXcolorrgb 28118 42964 a SDict begin H.S end 28118 42964 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 28768 42108 a SDict begin H.R end 28768 42108 a 28768 42964 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest 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begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.18) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 2751 -2672 a Ft(The)447 b(union)g(of)h(all)g(these)f(w)-36 b(alks)448 b(then)e(sho)-36 b(w)448 b(all)g(app)36 b(earing)447 b(pairs)g(in)g Fo(R)g Ft(con)-36 b(taining)448 b Fr(n)p Ft(,)j(and)446 b(their)800 -1067 y(cardinalit)-36 b(y)434 b(is)g(t)-36 b(wice)434 b(the)f(n)-36 b(um)g(b)36 b(er)432 b(of)j(in)-36 b(tegers)433 b Fr(k)414 b(<)369 b(n)434 b Ft(primes)f(with)h Fr(n)p Ft(,)g(so)g(the)f(lemma)h(is)g(pro)-36 b(v)g(ed.)p 51860 539 45 878 v 51905 -295 781 45 v 51905 539 V 52684 539 45 878 v 800 1535 a SDict begin H.S end 800 1535 a 800 1535 a SDict begin 14.5 H.A end 800 1535 a 800 1535 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.14) cvn H.B /DEST pdfmark end 800 1535 a Black 1716 x Fo(Lemma)684 b(14.)p Black 637 w Fs(L)-66 b(et)612 b Fr(n)644 b(>)f Ft(0)613 b Fs(b)-66 b(e)612 b(an)h(inte)-66 b(ger.)1040 b(Ther)-66 b(e)612 b(exists)h(an)g(inte)-66 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y(one,)495 b(that)490 b(is)f(if)g(the)g(value)h Fr(v)537 b Fs(app)-66 b(e)g(ars)490 b Fr(a)g Fs(times)e(in)h(the)g Fr(N)139 b Fs(-th)489 b(r)-66 b(ow,)496 b(then)489 b(it)g(app)-66 b(e)g(ars)489 b(at)h(le)-66 b(ast)490 b Fr(a)f Fs(times)800 12383 y(in)464 b(the)h Ft(\()p Fr(N)433 b Ft(+)295 b(3\))p Fs(-th)465 b(one.)2751 15095 y Ft(Both)415 b(of)h(the)f(t)-36 b(w)g(o)416 b(previous)g(lemmas)g(are)f(immediate)h(consequences)f(of)h (the)f(fact)h(that)f(eac)-36 b(h)416 b(w)-36 b(alk)800 16700 y(in)433 b(Lemma)p 0 .5 0 TeXcolorrgb 6960 16700 a SDict begin H.S end 6960 16700 a 0 .5 0 TeXcolorrgb Ft(13)p 0 .5 0 TeXcolorrgb 8261 15844 a SDict begin H.R end 8261 15844 a 8261 16700 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.13) cvn H.B /ANN pdfmark end 8261 16700 a Black 433 w Ft(sho)-36 b(ws)434 b(exactly)h(t)-36 b(w)g(o)434 b(no)36 b(des)433 b(b)36 b(et)-36 b(w)g(een)433 b(eac)-36 b(h)434 b(v)-72 b(alue)434 b Fr(n)p Ft(.)800 16958 y SDict begin H.S end 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/Subtype /Link /Dest (Prop.8) cvn H.B /ANN pdfmark end 12379 64076 a Black Ft(,)409 b(if)403 b(\()p Fr(w)15746 64275 y Fy(i)16122 64076 y Ft(\))16628 64275 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)20181 64076 y Ft(is)g(a)g(single-0)g(w)-36 b(alk,)411 b(then)402 b Fr(N)541 b Ft(is)403 b(of)g(the)f(form)h(3)p Fr(n)p Ft(,)410 b(where)403 b Fr(n)g Ft(is)g(an)800 65681 y(in)-36 b(teger.)800 65939 y SDict begin H.S end 800 65939 a 800 65939 a SDict begin 14.5 H.A end 800 65939 a 800 65939 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.16) cvn H.B /DEST pdfmark end 800 65939 a Black 26150 70521 a Ft(18)p Black eop %%Page: 19 19 19 18 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 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b(are)e(min)-36 b(us)483 b(signs,)800 23530 y(so)444 b(there)g(is)g(at)g(least)g(one)g(p)36 b(osition)445 b(in)e(the)h(w)-36 b(alk)445 b(where)f(w)-36 b(e)444 b(\257nd)f(the)g(succession)h(+)302 b Fq(\241)g(\241)p Ft(,)447 b(whic)-36 b(h)444 b(can)800 25135 y(b)36 b(e)483 b(remo)-36 b(v)g(ed)483 b(thanks)g(to)g(the)f(considerations)i(made)f (in)f(section)p 0 .5 0 TeXcolorrgb 34044 25135 a SDict begin H.S end 34044 25135 a 0 .5 0 TeXcolorrgb Ft(2.4)p 0 .5 0 TeXcolorrgb 35706 24279 a SDict begin H.R end 35706 24279 a 35706 25135 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.2.4) cvn H.B /ANN pdfmark end 35706 25135 a Black Ft(.)726 b(W)-108 b(e)483 b(then)g(get)g(a)g(single-0)h(w)-36 b(alk)800 26740 y(\()266 b(~)-916 b Fr(w)2236 26939 y Fy(i)2611 26740 y Ft(\))3117 26939 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)94 b Fw(\241)p Fv(3)7976 26740 y Ft(whic)-36 b(h,)523 b(b)-36 b(y)505 b(induction)g(h)-36 b(yp)36 b(othesis,)523 b(v)-36 b(eri\257es)506 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b(th)-36 b(us)390 b(ha)-36 b(v)g(e)391 b Fr(w)15396 33360 y Fy(N)16661 33161 y Ft(=)369 b Fq(\241)p Ft(,)400 b(and)390 b(there)g(m)-36 b(ust)391 b(b)36 b(e)390 b(exactly)j Fr(n)e Ft(plus)f(signs)i(and)e(2)p Fr(n)i Ft(min)-36 b(us)390 b(signs)800 34766 y(among)357 b(the)e Fr(w)8006 34965 y Fy(i)8382 34766 y Ft(s)h(for)h Fr(i)369 b Fq(2)f Ft([0)p Fr(;)221 b Ft(3)p Fr(n)137 b Fq(\241)g Ft(1].)556 b(Consequen)-36 b(tly)-108 b(,)372 b(the)356 b(sequence)g(+)137 b Fq(\241)g(\241)357 b Ft(m)-36 b(ust)355 b(app)36 b(ear)356 b(somewhere.)800 36371 y(Suppress)381 b(it:)553 b(the)382 b(cardinalit)-36 b(y)384 b(prop)36 b(ert)-36 b(y)382 b(remains)g(true)g(for)i(the)e(new)g(sequence,)394 b(whic)-36 b(h)382 b(is)h(therefore)g(a)800 37976 y(single-0)388 b(w)-36 b(alk)389 b(b)-36 b(y)388 b(the)f(induction)g(h)-36 b(yp)36 b(othesis.)563 b(Re-in)-36 b(tro)36 b(ducing)387 b(the)g(+)202 b Fq(\241)g(\241)387 b Ft(do)36 b(es)388 b(not)f(in)-36 b(tro)36 b(duce)387 b(one)800 39581 y(more)484 b(0)g(in)g(the)f(random)h(Fib)36 b(onacci)484 b(sequence,)497 b(so)484 b(the)f(full)i(sequence)e(\()p Fr(w)38862 39780 y Fy(i)39238 39581 y Ft(\))39744 39780 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)43378 39581 y Ft(is)i(also)f(a)h(single-0)800 41186 y(w)-36 b(alk,)435 b(and)e(the)g(prop)36 b(osition)434 b(is)g(pro)-36 b(v)g(ed.)p 51860 41186 45 878 v 51905 40352 781 45 v 51905 41186 V 52684 41186 45 878 v 800 42441 a SDict begin H.S end 800 42441 a 800 42441 a SDict begin 14.5 H.A end 800 42441 a 800 42441 a SDict begin [ /View [/XYZ H.V] /Dest (Cor.4) cvn H.B /DEST pdfmark end 800 42441 a Black 1457 x Fo(Corollary)521 b(4.)p Black 554 w Fs(A)464 b(single-)p Ft(0)g Fs(walk)i Ft(\()p Fr(w)19732 44097 y Fy(i)20107 43898 y Ft(\))20613 44097 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fv(3)p Fy(n)24429 43898 y Fs(c)-66 b(ontains)463 b Fr(n)j Fs(plus)g(signs)e(and)h Ft(2)p Fr(n)296 b Ft(+)f(1)465 b Fs(minus)f(signs.)2751 46610 y Ft(The)527 b(prop)36 b(ert)-36 b(y)527 b(men)-36 b(tioned)526 b(in)h(the)f(previous)i(prop)36 b(osition)527 b(has)g(for)g(consequence)g(the)g(follo)-36 b(wing)800 48215 y(\(see)434 b(the)f(sequence)p 0 1 0 0 TeXcolorcmyk 11039 48481 a SDict begin H.S end 11039 48481 a 0 1 0 0 TeXcolorcmyk -266 x Ft(A001764)p 11039 48428 4878 54 v 0 1 0 0 TeXcolorcmyk 15916 47307 a SDict begin H.R end 15916 47307 a 15916 48215 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A001764) >> /Subtype /Link H.B /ANN pdfmark end 15916 48215 a Black 434 w Ft(in)g([)p 0 .5 0 TeXcolorrgb 18228 48215 a SDict begin H.S end 18228 48215 a 0 .5 0 TeXcolorrgb Ft(7)p 0 .5 0 TeXcolorrgb 18879 47359 a SDict begin H.R end 18879 47359 a 18879 48215 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.EIT) cvn H.B /ANN pdfmark end 18879 48215 a Black Ft(])h(for)g(details)g(and)f(references\):)800 48547 y SDict begin H.S end 800 48547 a 800 48547 a SDict begin 14.5 H.A end 800 48547 a 800 48547 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.10) cvn H.B /DEST pdfmark end 800 48547 a Black 2380 x Fo(Prop)42 b(osition)500 b(10.)p Black 554 w Fs(L)-66 b(et)464 b(us)h(denote)f(by)h Fr(!)21928 51126 y Fy(n)23019 50927 y Fs(the)g(numb)-66 b(er)463 b(of)i(single-)p Ft(0)f Fs(walks)i Ft(\()p Fr(w)40996 51126 y Fy(i)41371 50927 y Ft(\))41877 51126 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fv(3)p Fy(n)45228 50927 y Fs(.)598 b(We)464 b(have)21363 54569 y Fr(!)22173 54768 y Fy(n)23168 54569 y Ft(=)26196 53670 y(1)p 24682 54264 3680 54 v 24682 55480 a(2)p Fr(n)295 b Ft(+)g(1)28493 52696 y Fm(\265)29471 53670 y Ft(3)p Fr(n)29796 55480 y(n)30898 52696 y Fm(\266)31876 54569 y Fr(:)2751 58045 y Ft(W)-108 b(e)351 b(can)f(deduce)g(from)h(what)f(precedes)g(the)g (follo)-36 b(wing)353 b(description)d(of)h(w)-36 b(alks)352 b(\()p Fr(w)43182 58244 y Fy(i)43557 58045 y Ft(\))44063 58244 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(N)47565 58045 y Ft(suc)-36 b(h)349 b(that)800 59650 y Fr(w)1730 59849 y Fy(N)3040 59650 y Ft(=)413 b(0)461 b(\(without)e(necessarily)i Fr(w)18387 59849 y Fy(i)19176 59650 y Fq(6)p Ft(=)414 b(0)460 b(for)g Fr(i)414 b(<)f(N)139 b Ft(\).)656 b(Let)460 b(us)f(giv)-36 b(e)461 b(it)f(brie\260y:)631 b(w)-36 b(e)460 b(denote)f(b)-36 b(y)460 b Fr(z)49955 59849 y Fv(1)50481 59650 y Ft(,)467 b Fr(z)51913 59849 y Fv(2)52439 59650 y Ft(,)800 61255 y Fr(z)1404 61454 y Fv(3)1930 61255 y Ft(.)217 b(.)g(.)g(the)342 b(increasing)g(v)-72 b(alues)343 b(b)36 b(et)-36 b(w)g(een)341 b(0)h(and)g Fr(N)480 b Ft(suc)-36 b(h)341 b(that)h Fr(g)31654 61454 y Fy(z)32092 61577 y Fl(1)32553 61454 y Fv(+2)34179 61255 y Ft(=)369 b Fr(g)36183 61454 y Fy(z)36621 61577 y Fl(2)37081 61454 y Fv(+2)38708 61255 y Ft(=)g Fr(g)40712 61454 y Fy(z)41150 61577 y Fl(3)41610 61454 y Fv(+2)43237 61255 y Ft(=)g Fq(\242)221 b(\242)g(\242)369 b Ft(=)g(0.)548 b(Then,)800 62860 y(the)406 b(w)-36 b(alks)408 b(de\257ned)d(b)-36 b(y)406 b(\()p Fr(w)14202 63059 y Fy(i)14578 62860 y Ft(\))15084 63059 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(z)17776 63182 y Fl(1)18292 62860 y Ft(,)412 b(\()p Fr(w)20501 63059 y Fy(i)20877 62860 y Ft(\))21383 63059 y Fy(z)21821 63182 y Fl(1)22281 63059 y Fv(+3)p Fw(\267)p Fy(i)p Fw(\267)p Fy(z)25705 63182 y Fl(2)26222 62860 y Ft(,)g(\()p Fr(w)28431 63059 y Fy(i)28806 62860 y Ft(\))29312 63059 y Fy(z)29750 63182 y Fl(2)30211 63059 y Fv(+3)p Fw(\267)p Fy(i)p Fw(\267)p Fy(z)33635 63182 y Fl(3)34152 62860 y Ft(,)g(.)217 b(.)g(.)g(are)407 b(single-0)g(w)-36 b(alks)408 b(\(and,)j(th)-36 b(us)800 64465 y(can)366 b(b)36 b(e)365 b(describ)36 b(ed)365 b(with)g(the)g(use)h(of)g(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 26092 64465 a SDict begin H.S end 26092 64465 a 0 .5 0 TeXcolorrgb Ft(9)p 0 .5 0 TeXcolorrgb 26742 63609 a SDict begin H.R end 26742 63609 a 26742 64465 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.9) cvn H.B /ANN pdfmark end 26742 64465 a Black Ft(\),)380 b(and)365 b(the)g(elemen)-36 b(ts)365 b Fr(w)38802 64664 y Fy(z)39240 64787 y Fl(1)39701 64664 y Fv(+1)40958 64465 y Ft(,)380 b Fr(w)42629 64664 y Fy(z)43067 64787 y Fl(1)43528 64664 y Fv(+2)44785 64465 y Ft(,)g Fr(w)46456 64664 y Fy(z)46894 64787 y Fl(2)47354 64664 y Fv(+1)48612 64465 y Ft(,)f Fr(w)50282 64664 y Fy(z)50720 64787 y Fl(2)51181 64664 y Fv(+2)52439 64465 y Ft(,)800 66070 y Fr(w)1730 66269 y Fy(z)2168 66392 y Fl(3)2629 66269 y Fv(+1)3886 66070 y Ft(,)434 b Fr(w)5611 66269 y Fy(z)6049 66392 y Fl(3)6510 66269 y Fv(+2)7767 66070 y Ft(.)217 b(.)g(.)g(are)434 b(allo)-36 b(w)g(ed)435 b(to)f(b)36 b(e)433 b(plus)g(or)h(min)-36 b(us)433 b(signs)h(without)f(condition.)p Black 26150 70521 a(19)p Black eop %%Page: 20 20 20 19 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.20) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.3) cvn H.B /DEST pdfmark end 800 -4000 a 1328 x Fn(4.3)1793 b(The)598 b(matrix)h(p)50 b(oin)-50 b(t)599 b(of)e(view)800 -209 y Ft(Let)463 b(us)g(k)-36 b(eep)463 b(the)g(no)36 b(des)463 b(and)g(edges)h(of)g Fo(R)f Ft(and)g(the)g(rules)g(for)h(the)f(v)-72 b(alues)464 b(at)f(eac)-36 b(h)464 b(no)36 b(de,)470 b(but)463 b(let)g(us)800 1396 y(replace)368 b(the)f(\257rst)g(app)36 b(earing)367 b(pair)h(\(1)p Fr(;)221 b Ft(1\))368 b(b)-36 b(y)368 b(formal)g(expressions)g Fr(a)g Ft(and)f Fr(b)p Ft(,)381 b(to)368 b(get)f(the)g(follo)-36 b(wing)370 b(tree:)p Black 21261 6539 a @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 9 10 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3149 x @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 8 9 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 x @beginspecial @setspecial 2.455 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 7 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.7825 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 7 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8147 15987 a @beginspecial @setspecial 2.455 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 26229 w @beginspecial @setspecial 8.24 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.325 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8147 19137 a @beginspecial @setspecial 0.86 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.05 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 18221 w @beginspecial @setspecial 8.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 16015 w @beginspecial @setspecial 11.74 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.91 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3124 22287 a @beginspecial @setspecial 0.86 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 10047 w @beginspecial @setspecial 3.08 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.02 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 13197 w @beginspecial @setspecial 7.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.18 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 11023 w @beginspecial @setspecial 11.74 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 9985 w @beginspecial @setspecial 13.92 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.9 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3124 25436 a @beginspecial @setspecial 0.24 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.48 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 3.08 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 4.4 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.64 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 7.24 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 8.56 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.8 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8063 w @beginspecial @setspecial 11.12 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.36 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6867 w @beginspecial @setspecial 13.92 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6236 w @beginspecial @setspecial 15.28 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.52 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 21261 6539 a @beginspecial @setspecial 0.356 1864679 mul 65536 div 72.27 div 72 mul 0.321 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3149 x @beginspecial @setspecial 0.315 1864679 mul 65536 div 72.27 div 72 mul 0.432 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3150 x @beginspecial @setspecial 1.041 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 8147 15987 a @beginspecial @setspecial 0.356 1864679 mul 65536 div 72.27 div 72 mul 0.321 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 26229 w @beginspecial @setspecial 1.247 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 8147 19137 a @beginspecial @setspecial 1.247 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 18221 w @beginspecial @setspecial 0.315 1864679 mul 65536 div 72.27 div 72 mul 0.432 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 16015 w @beginspecial @setspecial 1.453 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3124 22287 a @beginspecial @setspecial 1.041 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 10047 w @beginspecial @setspecial 1.247 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 13197 w @beginspecial @setspecial 1.247 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 11023 w @beginspecial @setspecial 1.041 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 9985 w @beginspecial @setspecial 1.453 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3124 25436 a @beginspecial @setspecial 1.453 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6992 w @beginspecial @setspecial 0.356 1864679 mul 65536 div 72.27 div 72 mul 0.321 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6110 w @beginspecial @setspecial 1.453 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6992 w @beginspecial @setspecial 1.247 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6110 w @beginspecial @setspecial 1.247 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 8063 w @beginspecial @setspecial 1.453 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6867 w @beginspecial @setspecial 1.247 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 6236 w @beginspecial @setspecial 1.453 1864679 mul 65536 div 72.27 div 72 mul 0.465 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 1171 28586 a @beginspecial @setspecial 1.081 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3905 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 5040 w @beginspecial @setspecial 1.081 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4157 w @beginspecial @setspecial 1.081 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3906 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 5039 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4158 w @beginspecial @setspecial 0.290 1864679 mul 65536 div 72.27 div 72 mul 0.384 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3905 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4158 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3905 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4914 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 4283 w @beginspecial @setspecial 1.257 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 3906 w @beginspecial @setspecial 1.433 1864679 mul 65536 div 72.27 div 72 mul 0.413 1864679 mul 65536 div 72.27 div 72 mul [ 1 1 1 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true psrectangle @endspecial 20920 6825 a Fr(a)20985 10149 y(b)19843 13248 y(a)295 b Ft(+)f Fr(b)7806 16273 y(a)32632 16398 y(a)h Ft(+)f(2)p Fr(b)6403 19547 y Ft(2)p Fr(a)i Ft(+)e Fr(b)26091 19598 y(b)40314 19547 y Ft(2)p Fr(a)i Ft(+)e(3)p Fr(b)1705 22697 y(a)h Ft(+)g Fr(b)6884 b Ft(3)p Fr(a)295 b Ft(+)g Fr(b)9709 b(a)295 b Ft(+)f(3)p Fr(b)7862 b(a)294 b Ft(+)h Fr(b)6496 b Ft(3)p Fr(a)296 b Ft(+)f(5)p Fr(b)1054 25847 y Ft(3)p Fr(a)h Ft(+)f(2)p Fr(b)9774 25722 y(a)14157 25847 y Ft(5)p Fr(a)g Ft(+)g(2)p Fr(b)3179 b(a)295 b Ft(+)g(2)p Fr(b)2622 b(a)295 b Ft(+)g(4)p Fr(b)4250 b Ft(3)p Fr(a)295 b Ft(+)g(4)p Fr(b)3053 b(a)295 b Ft(+)g(2)p Fr(b)2423 b Ft(5)p Fr(a)296 b Ft(+)f(8)p Fr(b)-312 28924 y Fj(2)p Fi(a)246 b Fj(+)g Fi(b)662 b Fj(4)p Fi(a)247 b Fj(+)f(3)p Fi(b)1797 b Fj(4)p Fi(a)246 b Fj(+)g Fi(b)1191 b Fj(2)p Fi(a)247 b Fj(+)f Fi(b)662 b Fj(8)p Fi(a)246 b Fj(+)g(3)p Fi(b)1520 b Fj(2)p Fi(a)247 b Fj(+)f(5)p Fi(b)27138 28970 y(b)29521 28924 y Fj(2)p Fi(a)h Fj(+)f(7)p Fi(b)638 b Fj(2)p Fi(a)246 b Fj(+)g(3)p Fi(b)386 b Fj(4)p Fi(a)247 b Fj(+)f(5)p Fi(b)1393 b Fj(4)p Fi(a)247 b Fj(+)f(7)p Fi(b)764 b Fj(2)p Fi(a)247 b Fj(+)e(3)p Fi(b)110 b Fj(8)p Fi(a)246 b Fj(+)g(13)p Fi(b)800 34885 y SDict begin H.S end 800 34885 a 800 34885 a SDict begin 14.5 H.A end 800 34885 a 800 34885 a SDict begin [ /View [/XYZ H.V] /Dest (figure.4) cvn H.B /DEST pdfmark end 800 34885 a Black 14651 37597 a Ft(Figure)433 b(4:)p 0 TeXcolorgray 20341 35992 a SDict begin H.S end 20341 35992 a 0 TeXcolorgray 0 TeXcolorgray 20341 35992 a SDict begin H.R end 20341 35992 a 20341 35992 a SDict begin [ /View [/XYZ H.V] /Dest (figure.4) cvn H.B /DEST pdfmark end 20341 35992 a Black 1605 x Ft(The)h(general)g(restricted)f(tree)g Fo(R)36999 37804 y Fv(\()p Fy(a;b)p Fv(\))p Black Black 2751 41180 a Ft(It)642 b(is)g(easily)h(seen)e(that,)694 b(for)642 b(an)-36 b(y)642 b(nonegativ)-36 b(e)642 b Fr(a)g Ft(and)f Fr(b)p Ft(,)694 b(all)642 b(the)g(recurrence)e(relations)i(giv)-36 b(en)800 42785 y(previously)575 b(concerning)f(sum)f(of)i(ro)-36 b(ws)574 b(remain)g(true,)609 b(up)573 b(to)h(a)g(c)-36 b(hange)574 b(in)g(initial)h(v)-72 b(alues)574 b(of)h(those)800 44390 y(sequences.)2751 45995 y(Lemma)p 0 .5 0 TeXcolorrgb 7251 45995 a SDict begin H.S end 7251 45995 a 0 .5 0 TeXcolorrgb Ft(2)p 0 .5 0 TeXcolorrgb 7901 45139 a SDict begin H.R end 7901 45139 a 7901 45995 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.2) cvn H.B /ANN pdfmark end 7901 45995 a Black 291 w Ft(lea)-36 b(v)g(es)292 b(to)f(the)g(follo)-36 b(wing)293 b(in)-36 b(terpretation)290 b(of)i(this)e(tree)h(in)g(terms) f(of)i(elemen)-36 b(ts)291 b(of)g(SL\(2)p Fr(;)221 b Fd(N)p Ft(\):)800 47600 y(the)501 b(no)36 b(des)500 b(of)i(the)f (general)g(restricted)g(tree)f(are)i(all)g(of)f(the)g(form)g Fr(ma)341 b Ft(+)g Fr(nb)p Ft(.)781 b(W)-108 b(e)501 b(deduce)f(from)h(this)800 49205 y(tree)455 b(another)g(tree)g(in)g (whic)-36 b(h)455 b(the)g(edge)g(joining)h(the)f(no)36 b(des)455 b Fr(ma)309 b Ft(+)h Fr(nb)455 b Ft(and)g Fr(m)40551 48723 y Fw(0)40861 49205 y Fr(a)310 b Ft(+)f Fr(n)43951 48723 y Fw(0)44262 49205 y Fr(b)456 b Ft(is)f(lab)36 b(elled)456 b(b)-36 b(y)800 51512 y(the)433 b(matrix)7267 49639 y Fm(\265)8954 50698 y Fr(m)1417 b(n)8799 52303 y(m)9937 51821 y Fw(0)11354 52303 y Fr(n)12130 51821 y Fw(0)12994 49639 y Fm(\266)13972 51512 y Ft(,)434 b(whic)-36 b(h)433 b(b)36 b(elongs)434 b(to)g(SL)o(\(2)p Fr(;)221 b Fd(Z)p Ft(\))434 b(b)-36 b(y)433 b(Lemma)p 0 .5 0 TeXcolorrgb 36294 51512 a SDict begin H.S end 36294 51512 a 0 .5 0 TeXcolorrgb Ft(2)p 0 .5 0 TeXcolorrgb 36944 50655 a SDict begin H.R end 36944 50655 a 36944 51512 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.2) cvn H.B /ANN pdfmark end 36944 51512 a Black Ft(.)p Black 26150 70521 a(20)p Black eop %%Page: 21 21 21 20 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.21) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 21261 -6299 a @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 8 10 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6299 x @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 6 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6299 x @beginspecial @setspecial 2.455 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 4 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.7825 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 4 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8147 12598 a @beginspecial @setspecial 2.455 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 2 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 26229 w @beginspecial @setspecial 8.24 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 2 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.325 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 2 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8147 18898 a @beginspecial @setspecial 0.66 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 0 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.05 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 0 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 18221 w @beginspecial @setspecial 8.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 0 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 16015 w @beginspecial @setspecial 11.52 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 0 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.91 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 0 2 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 2494 25197 a @beginspecial @setspecial 0.66 0.66 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 10677 w @beginspecial @setspecial 2.7 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.4 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 13197 w @beginspecial @setspecial 6.86 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.56 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 10330 w @beginspecial @setspecial 11.52 11.52 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 10678 w @beginspecial @setspecial 13.54 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.28 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul -2 0 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 21261 -6299 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6299 x @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6299 x @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8147 12598 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 26229 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8147 18898 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 18221 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 16015 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2494 25197 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 10677 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 13197 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 10330 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 10678 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2494 31496 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6425 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8504 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4598 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8504 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6173 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6363 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8630 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 21261 -3150 a @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 18526 -4690 a Fm(\265)20058 -3631 y Ft(1)1107 b(0)20058 -2026 y(0)g(1)23019 -4690 y Fm(\266)21261 3150 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 18526 1609 a(\265)20058 2668 y Ft(0)g(1)20058 4273 y(1)g(1)23019 1609 y Fm(\266)14704 9449 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 11969 7908 a(\265)13501 8967 y Ft(1)g(1)13501 10572 y(1)g(0)16462 7908 y Fm(\266)27819 9449 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 25083 7908 a(\265)26615 8967 y Ft(1)g(1)26615 10572 y(1)g(2)29576 7908 y Fm(\266)8147 15748 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 5412 14207 a(\265)6943 15266 y Ft(1)h(0)6943 16872 y(2)g(1)9905 14207 y Fm(\266)30372 15748 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 27637 14207 a(\265)29168 15266 y Ft(1)f(2)29168 16872 y(0)g(1)32129 14207 y Fm(\266)38380 15748 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 35644 14207 a(\265)37176 15266 y Ft(1)g(2)37176 16872 y(2)g(3)40137 14207 y Fm(\266)5321 22047 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 2585 20506 a(\265)4117 21566 y Ft(2)g(1)4117 23171 y(1)g(1)7078 20506 y Fm(\266)10659 22047 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 7924 20506 a(\265)9455 21566 y Ft(2)h(1)9455 23171 y(3)g(1)12416 20506 y Fm(\266)26368 22047 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 23633 20506 a(\265)25164 21566 y Ft(0)f(1)25164 23171 y(1)g(3)28125 20506 y Fm(\266)39541 22047 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 36806 20506 a(\265)38337 21566 y Ft(2)g(3)38337 23171 y(1)g(1)41298 20506 y Fm(\266)44880 22047 y @beginspecial @setspecial 3425047 65536 div 2087021 65536 div psdeletebox @endspecial 42144 20506 a(\265)43676 21566 y Ft(2)g(3)43676 23171 y(3)g(5)46637 20506 y Fm(\266)2494 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 419 27582 a Fc(\265)1534 28114 y Fl(1)g(1)1534 28889 y(3)g(2)4007 27582 y Fc(\266)11045 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 8970 27582 a(\265)10086 28114 y Fl(3)f(1)10086 28889 y(1)g(0)12558 27582 y Fc(\266)15297 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 13222 27582 a(\265)14338 28114 y Fl(3)g(1)14338 28889 y(5)g(2)16810 27582 y Fc(\266)24195 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 22119 27582 a(\265)23235 28114 y Fl(1)h(3)23235 28889 y(1)g(2)25707 27582 y Fc(\266)28446 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 26371 27582 a(\265)27487 28114 y Fl(1)g(3)27487 28889 y(1)g(4)29959 27582 y Fc(\266)36698 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 34623 27582 a(\265)35739 28114 y Fl(1)g(1)35739 28889 y(3)g(4)38211 27582 y Fc(\266)45218 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 43143 27582 a(\265)44259 28114 y Fl(3)g(5)44259 28889 y(1)g(2)46731 27582 y Fc(\266)49533 28346 y @beginspecial @setspecial 2643685 65536 div 1130206 65536 div psdeletebox @endspecial 47458 27582 a(\265)48574 28114 y Fl(3)g(5)48574 28889 y(5)g(8)51046 27582 y Fc(\266)800 31496 y SDict begin H.S end 800 31496 a 800 31496 a SDict begin 14.5 H.A end 800 31496 a 800 31496 a SDict begin [ /View [/XYZ H.V] /Dest (figure.5) cvn H.B /DEST pdfmark end 800 31496 a Black 18948 34208 a Ft(Figure)433 b(5:)p 0 TeXcolorgray 24638 32603 a SDict begin H.S end 24638 32603 a 0 TeXcolorgray 0 TeXcolorgray 24638 32603 a SDict begin H.R end 24638 32603 a 24638 32603 a SDict begin [ /View [/XYZ H.V] /Dest (figure.5) cvn H.B /DEST pdfmark end 24638 32603 a Black 1605 x Ft(The)h(SL)o(\(2)p Fr(;)221 b Fd(N)p Ft(\))434 b(tree)p Black Black 800 36090 a SDict begin H.S end 800 36090 a 800 36090 a SDict begin 14.5 H.A end 800 36090 a 800 36090 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.11) cvn H.B /DEST pdfmark end 800 36090 a Black 1605 x Fo(Prop)42 b(osition)647 b(11.)p Black 620 w Fs(The)582 b(latter)g(tr)-66 b(e)g(e)582 b(is)g(c)-66 b(omp)g(ose)g(d)582 b(of)h(al)66 b(l)583 b(elements)f(of)g Ft(SL)o(\(2)p Fr(;)221 b Fd(N)p Ft(\))p Fs(,)613 b(e)-66 b(ach)583 b(app)-66 b(e)g(aring)800 39300 y(exactly)464 b(onc)-66 b(e.)p Black 800 42012 a(Pr)g(o)g(of.)p Black 649 w Ft(The)558 b(fact)h(that)e(the)h(co)36 b(e\261cien)-36 b(ts)558 b(of)h(all)f(the)g(matrices)g(are)g (nonnegativ)-36 b(e)559 b(is)f(a)g(consequence)800 43617 y(of)473 b(Lemma)p 0 .5 0 TeXcolorrgb 7002 43617 a SDict begin H.S end 7002 43617 a 0 .5 0 TeXcolorrgb Ft(2)p 0 .5 0 TeXcolorrgb 7652 42761 a SDict begin H.R end 7652 42761 a 7652 43617 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.2) cvn H.B /ANN pdfmark end 7652 43617 a Black Ft(.)695 b(The)472 b(fact)h(that)f(eac)-36 b(h)473 b(matrice)g(cannot)f (app)36 b(ear)472 b(t)-36 b(wice)473 b(is)g(a)f(consequence)h(of)g(the) f(non-)800 45923 y(redundance)580 b(prop)36 b(ert)-36 b(y)582 b(of)h Fo(R)p Ft(.)1023 b(Finally)-108 b(,)620 b(if)23679 44050 y Fm(\265)25365 45110 y Fr(m)1418 b(n)25210 46715 y(m)26348 46233 y Fw(0)27765 46715 y Fr(n)28541 46233 y Fw(0)29406 44050 y Fm(\266)31005 45923 y Fq(2)621 b Ft(SL\(2)p Fr(;)221 b Fd(N)p Ft(\),)620 b(then)581 b(the)h(c)-36 b(haracterization)800 49244 y(of)520 b(w)-36 b(alks)520 b(in)f Fo(R)g Ft(\(Corollary)p 0 .5 0 TeXcolorrgb 15572 49244 a SDict begin H.S end 15572 49244 a 0 .5 0 TeXcolorrgb Ft(2)p 0 .5 0 TeXcolorrgb 16223 48388 a SDict begin H.R end 16223 48388 a 16223 49244 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Cor.2) cvn H.B /ANN pdfmark end 16223 49244 a Black Ft(\))g(suggests)g(to)g(consider)g(the)f (matrix)35877 47371 y Fm(\265)37408 48431 y Fq(j)p Fr(m)295 b Fq(\241)g Fr(m)41676 47949 y Fw(0)41986 48431 y Fq(j)1107 b(j)p Fr(n)296 b Fq(\241)f Fr(n)47007 47949 y Fw(0)47318 48431 y Fq(j)39313 50036 y Fr(m)4736 b(n)48241 47371 y Fm(\266)49738 49244 y Ft(as)519 b(its)800 51550 y(paren)-36 b(t.)748 b(Pursuing)490 b(in)g(the)g(same)h(w)-36 b(a)g(y)-108 b(,)505 b(b)-36 b(y)491 b(induction)f(w)-36 b(e)490 b(construct)g(an)g (e\256ectiv)-36 b(e)491 b(w)-36 b(alk)492 b(in)e(the)g(tree)800 53156 y(whic)-36 b(h)433 b(leads)h(to)g(the)f(desired)g(matrix,)i(and)e (the)g(prop)36 b(osition)433 b(is)h(pro)-36 b(v)g(ed.)p 51860 53156 45 878 v 51905 52322 781 45 v 51905 53156 V 52684 53156 45 878 v 800 54410 a SDict begin H.S end 800 54410 a 800 54410 a SDict begin 14.5 H.A end 800 54410 a 800 54410 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.12) cvn H.B /DEST pdfmark end 800 54410 a Black 1458 x Fo(Prop)42 b(osition)499 b(12.)p Black 553 w Fs(If)463 b Fr(M)508 b Fq(2)369 b Ft(SL)o(\(2)p Fr(;)221 b Fd(N)p Ft(\))465 b Fs(is)f(attaine)-66 b(d)463 b(by)h(the)f(walk)i Ft(\()p Fr(w)35505 56067 y Fy(i)35880 55868 y Ft(\))36386 56067 y Fv(0)p Fw(\267)p Fy(i)p Fw(\267)p Fy(n)39731 55868 y Fs(in)e(the)h(tr)-66 b(e)g(e,)462 b(then)h Fr(M)50672 55386 y Fy(T)51870 55868 y Fs(is)800 57473 y(attaine)-66 b(d)445 b(by)i(the)g(walk)g Ft(\()p 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72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.325 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8147 10998 a @beginspecial @setspecial 0.86 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.05 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 18221 w @beginspecial @setspecial 8.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 16015 w @beginspecial @setspecial 11.74 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.91 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3124 14148 a @beginspecial @setspecial 0.86 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 10047 w @beginspecial @setspecial 3.08 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.02 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 13197 w @beginspecial @setspecial 7.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.18 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 11023 w @beginspecial @setspecial 11.74 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 9985 w @beginspecial @setspecial 13.92 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.9 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3124 17298 a @beginspecial @setspecial 0.24 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.48 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 3.08 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 4.4 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.64 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 7.24 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 8.56 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.8 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8063 w @beginspecial @setspecial 11.12 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.36 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6867 w @beginspecial @setspecial 13.92 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6236 w @beginspecial @setspecial 15.28 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.52 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1171 20447 a @beginspecial @setspecial 0.24 0.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 1.12 1.48 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.84 1.48 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 5040 w @beginspecial @setspecial 2.72 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 3.44 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4157 w @beginspecial @setspecial 4.4 4.4 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3906 w @beginspecial @setspecial 5.28 5.64 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 6 5.64 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 5039 w @beginspecial @setspecial 6.88 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 7.6 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4158 w @beginspecial @setspecial 8.56 8.56 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 9.44 9.8 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.16 9.8 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4158 w @beginspecial @setspecial 11.12 11.12 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 12 12.36 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.72 12.36 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4914 w @beginspecial @setspecial 13.6 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.32 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4283 w @beginspecial @setspecial 15.28 15.28 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3906 w @beginspecial @setspecial 16.16 16.52 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.88 16.52 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 21261 -1600 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3150 x @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3149 x @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8147 7849 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 26229 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8147 10998 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 18221 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 16015 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3124 14148 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 10047 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 13197 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 11023 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 9985 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3124 17298 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6992 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6110 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6992 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6110 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8063 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6867 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6236 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 1171 20447 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3905 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 5040 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4157 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3906 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 5039 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4158 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3905 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4158 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3905 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4914 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4283 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3906 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 1171 23597 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2267 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3023 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2771 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3024 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2771 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3024 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2771 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2267 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3024 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2267 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 21261 -25 a @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 20756 307 a Ft(+)21261 3124 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 21045 3410 a(-)14704 6274 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 14488 6560 a(-)27819 6274 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 27313 6606 a(+)8147 9424 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 7641 9756 a(+)30372 9424 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 29866 9756 a(+)38380 9424 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 38163 9710 a(-)5635 12573 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 5130 12905 a(+)10659 12573 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 10442 12859 a(-)26368 12573 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 26151 12859 a(-)39887 12573 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 39671 12859 a(-)44880 12573 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 44374 12905 a(+)3124 15723 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 2907 16009 a(-)11643 15723 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 11427 16009 a(-)14698 15723 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 14193 16055 a(+)24793 15723 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 24576 16009 a(-)27848 15723 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 27342 16055 a(+)37391 15723 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 36886 16055 a(+)45817 15723 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 45311 16055 a(+)48935 15723 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 48718 16009 a(-)2147 18872 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 1931 19158 a(-)4100 18872 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 3594 19205 a(+)10116 18872 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 9610 19205 a(+)15250 18872 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 14744 19205 a(+)17202 18872 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 16986 19158 a(-)23218 18872 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 22712 19205 a(+)28352 18872 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 27846 19205 a(+)30305 18872 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 30088 19158 a(-)36415 18872 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 35909 19205 a(+)38368 18872 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 38151 19158 a(-)44257 18872 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 44041 19158 a(-)49517 18872 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 49301 19158 a(-)51470 18872 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 50964 19205 a(+)1171 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 665 22354 a(+)4509 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 4004 22354 a(+)5643 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 5427 22308 a(-)9549 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 9043 22354 a(+)10683 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 10466 22308 a(-)14273 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 14056 22308 a(-)17612 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 17395 22308 a(-)18746 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 18240 22354 a(+)22651 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 22145 22354 a(+)23785 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 23568 22308 a(-)27376 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 27159 22308 a(-)30714 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 30497 22308 a(-)31848 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 31342 22354 a(+)35439 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 35222 22308 a(-)38777 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 38560 22308 a(-)39911 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 39405 22354 a(+)43754 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 43537 22308 a(-)44887 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 44382 22354 a(+)48541 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 48035 22354 a(+)51880 22022 y @beginspecial @setspecial 785399 65536 div 699839 65536 div psdeletebox @endspecial 51374 22354 a(+)53013 22022 y @beginspecial @setspecial 443159 65536 div 525082 65536 div psdeletebox @endspecial 52797 22308 a(-)800 29896 y SDict begin H.S end 800 29896 a 800 29896 a SDict begin 14.5 H.A end 800 29896 a 800 29896 a SDict begin [ /View [/XYZ H.V] /Dest (figure.6) cvn H.B /DEST pdfmark end 800 29896 a Black 17611 32608 a Ft(Figure)433 b(6:)p 0 TeXcolorgray 23301 31003 a SDict begin H.S end 23301 31003 a 0 TeXcolorgray 0 TeXcolorgray 23301 31003 a SDict begin H.R end 23301 31003 a 23301 31003 a SDict begin [ /View [/XYZ H.V] /Dest (figure.6) cvn H.B /DEST pdfmark end 23301 31003 a Black 1605 x Ft(The)g(determinan)-36 b(ts)433 b(tree)p Black Black 2751 36021 a(The)326 b(structure)e(of)j (the)e(plus)h(and)f(min)-36 b(us)325 b(signs)h(is)g(giv)-36 b(en)327 b(b)-36 b(y)325 b(the)h(follo)-36 b(wing)328 b(rules,)347 b(routinely)326 b(pro)-36 b(v)g(ed:)800 36280 y SDict begin H.S end 800 36280 a 800 36280 a SDict begin 14.5 H.A end 800 36280 a 800 36280 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.13) cvn H.B /DEST pdfmark end 800 36280 a Black 2453 x Fo(Prop)42 b(osition)668 b(13.)p Black 629 w Fs(L)-66 b(et)598 b Fr(")619 b Fq(2)f(f\241)p Fr(;)221 b Ft(+)p Fq(g)600 b Fs(b)-66 b(e)599 b(the)g(lab)-66 b(el)599 b(of)g(an)h(e)-66 b(dge.)1000 b(The)600 b(right)e(e)-66 b(dge)599 b(fol)66 b(lowing)599 b(it)g(is)800 40338 y(lab)-66 b(el)66 b(le)-66 b(d)464 b Fq(\241)p Fr(")h Fs(and)g(the)g(left)f(one)g (\(if)h(it)f(exists\))g(by)h Fr(")p Fs(.)2751 43050 y Ft(As)434 b(a)g(corollary)-108 b(,)435 b(w)-36 b(e)434 b(get)f(the)g(n)-36 b(um)g(b)36 b(er)432 b(of)j(+)e(and)g Fq(\241)h Ft(in)f(eac)-36 b(h)434 b(ro)-36 b(w)434 b(of)g(the)f(tree:) 800 43309 y SDict begin H.S end 800 43309 a 800 43309 a SDict begin 14.5 H.A end 800 43309 a 800 43309 a SDict begin [ /View [/XYZ H.V] /Dest (Cor.5) cvn H.B /DEST pdfmark end 800 43309 a Black 2453 x Fo(Corollary)703 b(5.)p Black 635 w Fs(The)611 b(numb)-66 b(er)610 b(of)h(plus)h(signs)e(at)h(the)g Fr(n)p Fs(-th)g(r)-66 b(ow)611 b(is)g Fr(F)36893 45961 y Fy(n)37519 45762 y Fr(=)p Ft(2)h Fs(if)e Fr(n)i Fs(is)e(of)h(the)g (form)f Ft(3)p Fr(k)45 b Fs(,)800 47368 y Fq(b)p Fr(F)2232 47567 y Fy(n)2859 47368 y Fr(=)p Ft(2)p Fq(c)296 b Ft(+)f(1)465 b Fs(if)f Fr(n)h Fs(is)g(of)g(the)f(form)g Ft(3)p Fr(k)341 b Ft(+)295 b(1)p Fs(,)465 b(and)g Fq(b)p Fr(F)26633 47567 y Fy(n)27259 47368 y Fr(=)p Ft(2)p Fq(c)h Fs(if)e Fr(n)h Fs(is)g(of)g(the)f(form)h Ft(3)p Fr(k)340 b Ft(+)295 b(2)p Fs(.)p Black 800 50080 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(Let)485 b(us)f(consider)h(the)g Fr(n)p Ft(-th)f(ro)-36 b(w)486 b(of)f(the)g(tree;)511 b(w)-36 b(e)485 b(denote)g(b)-36 b(y)485 b Fr(d)37458 49598 y Fv(+)37458 50408 y Fy(n)38730 50080 y Ft(\(resp.)732 b Fr(d)43325 49598 y Fw(\241)43325 50408 y Fy(n)44112 50080 y Ft(\))485 b(its)g(n)-36 b(um)g(b)36 b(er)483 b(of)800 51685 y(righ)-36 b(t)480 b(edges)h(lab)36 b(elled)480 b(+)g(\(resp.)718 b Fq(\241)p Ft(\))481 b(and)f(b)-36 b(y)480 b Fr(g)24881 51203 y Fv(+)24833 52013 y Fy(n)26148 51685 y Ft(\(resp.)718 b Fr(g)30724 51203 y Fw(\241)30676 52013 y Fy(n)31510 51685 y Ft(\))480 b(the)g(n)-36 b(um)g(b)36 b(er)479 b(of)i(left)g(edges)f(lab)36 b(elled)481 b(+)800 53290 y(\(resp.)578 b Fq(\241)p Ft(\).)g(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 14041 53290 a SDict begin H.S end 14041 53290 a 0 .5 0 TeXcolorrgb Ft(4)p 0 .5 0 TeXcolorrgb 14691 52434 a SDict begin H.R end 14691 52434 a 14691 53290 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.4) cvn H.B /ANN pdfmark end 14691 53290 a Black 434 w Ft(easily)435 b(giv)-36 b(es)435 b(the)e(follo)-36 b(wing)435 b(relations)g(for)f (all)g Fr(n)p Ft(:)15328 56223 y Fr(d)16004 55675 y Fv(+)16004 56552 y Fy(n)17086 56223 y Ft(+)295 b Fr(d)19069 55675 y Fw(\241)19069 56552 y Fy(n)20225 56223 y Ft(=)369 b Fr(F)22448 56422 y Fy(n)p Fw(\241)p Fv(1)25577 56223 y Ft(and)1300 b Fr(g)29644 55675 y Fv(+)29596 56552 y Fy(n)30726 56223 y Ft(+)295 b Fr(g)32704 55675 y Fw(\241)32656 56552 y Fy(n)33859 56223 y Ft(=)369 b Fr(F)36082 56422 y Fy(n)p Fw(\241)p Fv(2)37911 56223 y Fr(:)2751 59157 y Ft(The)434 b(structure)e(of)i(the)f(tree,)h(together)f(with)h(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 32771 59157 a SDict begin H.S end 32771 59157 a 0 .5 0 TeXcolorrgb Ft(13)p 0 .5 0 TeXcolorrgb 34072 58301 a SDict begin H.R end 34072 58301 a 34072 59157 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.13) cvn H.B /ANN pdfmark end 34072 59157 a Black Ft(,)433 b(also)i(giv)-36 b(e)435 b(the)e(follo)-36 b(wing:)17192 62090 y Fr(d)17868 61542 y Fv(+)17868 62419 y Fy(n)p Fv(+1)20066 62090 y Ft(=)368 b Fr(g)22117 61542 y Fw(\241)22069 62419 y Fy(n)23199 62090 y Ft(+)295 b Fr(d)25182 61542 y Fw(\241)25182 62419 y Fy(n)27270 62090 y Fr(d)27946 61542 y Fw(\241)27946 62419 y Fy(n)p Fv(+1)30143 62090 y Ft(=)369 b Fr(g)32195 61542 y Fv(+)32147 62419 y Fy(n)33277 62090 y Ft(+)294 b Fr(d)35259 61542 y Fv(+)35259 62419 y Fy(n)36046 62090 y Fr(;)20305 65024 y(g)20976 64475 y Fv(+)20928 65352 y Fy(n)p Fv(+1)23125 65024 y Ft(=)369 b Fr(d)25182 64475 y Fv(+)25182 65352 y Fy(n)27270 65024 y Fr(g)27941 64475 y Fw(\241)27893 65352 y Fy(n)p Fv(+1)30090 65024 y Ft(=)g Fr(d)32147 64475 y Fw(\241)32147 65352 y Fy(n)32934 65024 y Fr(:)p Black 26150 70521 a Ft(23)p Black eop %%Page: 24 24 24 23 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.24) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 2751 -2672 a Ft(W)-108 b(e)496 b(deduce)g(from)h(those)f(relations)h(that)f Fr(d)24954 -3221 y Fv(+)24954 -2346 y Fy(n)p Fv(+1)27120 -2672 y Ft(+)338 b Fr(g)29141 -3221 y Fv(+)29093 -2346 y Fy(n)p Fv(+1)31397 -2672 y Ft(=)475 b Fr(F)33726 -2473 y Fy(n)p Fw(\241)p Fv(1)35893 -2672 y Ft(+)337 b Fr(g)37913 -3154 y Fw(\241)37865 -2343 y Fy(n)39196 -2672 y Ft(and)496 b(that)g Fr(d)45345 -3221 y Fw(\241)45345 -2346 y Fy(n)p Fv(+1)47512 -2672 y Ft(+)337 b Fr(g)49532 -3221 y Fw(\241)49484 -2346 y Fy(n)p Fv(+1)51788 -2672 y Ft(=)800 -1067 y Fr(F)1642 -868 y Fy(n)p Fw(\241)p Fv(1)3766 -1067 y Ft(+)295 b Fr(g)5744 -1549 y Fv(+)5696 -738 y Fy(n)6530 -1067 y Ft(.)579 b(If)434 b(w)-36 b(e)434 b(denote)f(b)-36 b(y)433 b Fr(e)17285 -868 y Fy(n)18345 -1067 y Ft(the)g(di\256erence)g Fr(g)27073 -1549 y Fv(+)27025 -738 y Fy(n)28155 -1067 y Fq(\241)295 b Fr(g)30154 -1549 y Fw(\241)30106 -738 y Fy(n)30941 -1067 y Ft(,)433 b(then)g(w)-36 b(e)434 b(get)13047 1867 y Fr(e)13650 2066 y Fy(n)14645 1867 y Ft(=)369 b Fr(g)16697 1318 y Fv(+)16649 2195 y Fy(n)17779 1867 y Fq(\241)295 b Fr(g)19778 1318 y Fw(\241)19730 2195 y Fy(n)21672 1867 y Ft(=)1106 b Fr(d)24466 1318 y Fv(+)24466 2195 y Fy(n)p Fw(\241)p Fv(1)26590 1867 y Fq(\241)295 b Fr(d)28594 1318 y Fw(\241)28594 2195 y Fy(n)p Fw(\241)p Fv(1)21672 3804 y Ft(=)1106 b(\()p Fr(g)24967 3256 y Fw(\241)24919 4132 y Fy(n)p Fw(\241)p Fv(2)27043 3804 y Ft(+)295 b Fr(d)29026 3256 y Fw(\241)29026 4132 y Fy(n)p Fw(\241)p Fv(2)30854 3804 y Ft(\))g Fq(\241)g Ft(\()p Fr(g)34160 3256 y Fv(+)34112 4132 y Fy(n)p Fw(\241)p Fv(2)36236 3804 y Ft(+)g Fr(d)38219 3256 y Fv(+)38219 4132 y Fy(n)p Fw(\241)p Fv(2)40047 3804 y Ft(\))21672 5741 y(=)1106 b Fq(\241)p Fr(e)25426 5940 y Fy(n)p Fw(\241)p Fv(2)27550 5741 y Fq(\241)295 b Ft(\()p Fr(d)30060 5193 y Fv(+)30060 6070 y Fy(n)p Fw(\241)p Fv(2)32184 5741 y Fq(\241)g Fr(d)34188 5193 y Fw(\241)34188 6070 y Fy(n)p Fw(\241)p Fv(2)36017 5741 y Ft(\))21672 7678 y(=)1106 b Fq(\241)p Fr(e)25426 7877 y Fy(n)p Fw(\241)p Fv(2)27550 7678 y Fq(\241)295 b Ft(\()p Fr(g)30055 7130 y Fv(+)30007 8007 y Fy(n)p Fw(\241)p Fv(1)32131 7678 y Fq(\241)g Fr(g)34130 7130 y Fw(\241)34082 8007 y Fy(n)p Fw(\241)p Fv(1)35911 7678 y Ft(\))21672 9616 y(=)1106 b Fq(\241)p Ft(\()p Fr(e)25932 9815 y Fy(n)p Fw(\241)p Fv(2)28056 9616 y Ft(+)295 b Fr(e)29966 9815 y Fy(n)p Fw(\241)p Fv(1)31794 9616 y Ft(\))p Fr(:)2751 12549 y Ft(Since)562 b(an)g(immediate)g (computation)f(sho)-36 b(ws)563 b(that)e Fr(e)29827 12748 y Fv(3)30940 12549 y Ft(=)588 b Fq(\241)p Ft(1)562 b(and)f Fr(e)38045 12748 y Fv(4)39159 12549 y Ft(=)587 b(0,)595 b(w)-36 b(e)562 b(easily)h(get)f(that)800 14154 y(the)578 b(sequence)h(\()p Fr(e)9830 14353 y Fy(n)10456 14154 y Ft(\))10962 14353 y Fy(n)p Fw(\270)p Fv(3)13370 14154 y Ft(is)g(simply)h(the)e(sequence)h Fq(\241)p Ft(1,)616 b(0,)g(1,)g Fq(\241)p Ft(1,)g(0,)g(1,)g(etc.,)g(and)578 b(the)h(conclusion)800 15759 y(follo)-36 b(ws.)p 51860 15759 45 878 v 51905 14926 781 45 v 51905 15759 V 52684 15759 45 878 v 2751 18361 a(An)409 b(immediate)h(consequence)g(of)h (the)e(rules)h(giv)-36 b(en)410 b(in)g(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 37116 18361 a SDict begin H.S end 37116 18361 a 0 .5 0 TeXcolorrgb Ft(13)p 0 .5 0 TeXcolorrgb 38417 17505 a SDict begin H.R end 38417 17505 a 38417 18361 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.13) cvn H.B /ANN pdfmark end 38417 18361 a Black 409 w Ft(is)410 b(that)g(the)f(in\257nite)g(w)-36 b(ords)800 19966 y(in)525 b Fq(f\241)p Fr(;)221 b Ft(+)p Fq(g)527 b Ft(giv)-36 b(en)526 b(b)-36 b(y)525 b(w)-36 b(alks)527 b(in)f(the)f(determinan)-36 b(t)524 b(tree)h(are)h(exactly)h(those)e (whic)-36 b(h)525 b(nev)-36 b(er)526 b(sho)-36 b(w)526 b(the)800 21571 y(sequence)434 b Fq(\241)295 b(\241)g(\241)434 b Ft(neither)f(+)295 b(+)f(+.)578 b(In)434 b(particular:)800 21829 y SDict begin H.S end 800 21829 a 800 21829 a SDict begin 14.5 H.A end 800 21829 a 800 21829 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.14) cvn H.B /DEST pdfmark end 800 21829 a Black 2454 x Fo(Prop)42 b(osition)641 b(14.)p Black 617 w Fs(In)578 b Fq(f\241)p Fr(;)221 b Ft(+)p Fq(g)17306 23801 y Fp(N)18580 24283 y Fs(e)-66 b(quipp)g(e)g(d)577 b(with)h(the)g(cylinder)f(top)-66 b(olo)g(gy,)605 b(let)578 b(us)h(denote)f(by)g Fq(D)614 b Fs(the)800 25888 y(subset)456 b(of)g(wor)-66 b(ds)456 b(who)h(do)-66 b(es)456 b(not)f(show)i(anywher) -66 b(e)456 b(the)f(suc)-66 b(c)g(ession)455 b Fq(\241)275 b(\241)h(\241)456 b Fs(or)g Ft(+)274 b(+)h(+)p Fs(,)457 b(and)f(by)f Fq(W)565 b Fs(the)800 27493 y(subset)542 b(of)f(al)66 b(l)543 b(wor)-66 b(ds)542 b(which)g(never)f(show)h(the)g (suc)-66 b(c)g(ession)541 b Fq(\241\241)p Fs(.)827 b(T)-100 b(o)543 b(any)e Fr(w)547 b Fq(2)510 b(W)651 b Fs(we)542 b(asso)-66 b(ciate)541 b(its)800 29098 y(c)-66 b(orr)g(esp)g(onding)468 b(walk)i(in)f Fo(R)p Fs(,)i(then)e(make)g(the)h(c)-66 b(o)g(dage)469 b Fr(d)378 b Fq(2)g(D)506 b Fs(c)-66 b(orr)g(esp)g (onding)468 b(to)i(the)f(determinants)f(of)800 30703 y(the)500 b(e)-66 b(dges)500 b(of)f Fo(R)p Fs(.)703 b(Thus,)509 b(the)500 b(de\257ne)-66 b(d)499 b(function)g Fr(\301)h Fs(:)f Fq(W)544 b(\241)-221 b(!)434 b(D)537 b Fs(is)500 b(bije)-66 b(ctive.)700 b(Mor)-66 b(e)g(over,)507 b(the)500 b Fr(n)g Fs(\257rst)800 32308 y(letters)494 b(of)g Fr(\301)p Ft(\()p Fr(w)36 b Ft(\))494 b Fs(dep)-66 b(ends)494 b(only)g(on)h(the)f Fr(n)h Fs(\257rst)f(letters)g(of)g Fr(w)36 b Fs(,)501 b(and)495 b(c)-66 b(onversely)493 b(the)h Fr(n)h Fs(\257rst)f(letters)g (of)800 33913 y Fr(d)465 b Fs(dep)-66 b(ends)464 b(only)h(on)g(those)g (of)g Fr(\301)17124 33431 y Fw(\241)p Fv(1)18381 33913 y Ft(\()p Fr(d)p Ft(\))p Fs(.)597 b(In)464 b(p)-66 b(articular,)464 b Fr(\301)h Fs(is)f(bic)-66 b(ontinuous.)2751 36625 y Ft(Finally)-108 b(,)435 b(the)e(next)g(tree)g(is)h(obtained)g(b)-36 b(y)433 b(replacing)h(eac)-36 b(h)433 b(matrix)i(b)-36 b(y)433 b(its)h(trace.)p Black 26150 70521 a(24)p Black eop %%Page: 25 25 25 24 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.25) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 19661 -850 a @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 9 10 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3149 x @beginspecial @setspecial 6.61875 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 8 9 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3150 x @beginspecial @setspecial 2.455 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 7 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.7825 6.61875 sub 1864679 mul 65536 div 72.27 div 72 mul 7 8 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6547 8598 a @beginspecial @setspecial 2.455 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 26229 w @beginspecial @setspecial 8.24 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 13.325 10.7825 sub 1864679 mul 65536 div 72.27 div 72 mul 6 7 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6547 11748 a @beginspecial @setspecial 0.86 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 4.05 2.455 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 18221 w @beginspecial @setspecial 8.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 16016 w @beginspecial @setspecial 11.74 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.91 13.325 sub 1864679 mul 65536 div 72.27 div 72 mul 5 6 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1524 14898 a @beginspecial @setspecial 0.86 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 10047 w @beginspecial @setspecial 3.08 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.02 4.05 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 13197 w @beginspecial @setspecial 7.24 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.18 8.24 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 11023 w @beginspecial @setspecial 11.74 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 9985 w @beginspecial @setspecial 13.92 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 15.9 14.91 sub 1864679 mul 65536 div 72.27 div 72 mul 4 5 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 1524 18047 a @beginspecial @setspecial 0.24 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.48 0.86 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 3.08 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 4.4 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 5.64 5.02 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6992 w @beginspecial @setspecial 7.24 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6110 w @beginspecial @setspecial 8.56 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 9.8 9.18 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 8063 w @beginspecial @setspecial 11.12 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.36 11.74 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6867 w @beginspecial @setspecial 13.92 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 6236 w @beginspecial @setspecial 15.28 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 16.52 15.9 sub 1864679 mul 65536 div 72.27 div 72 mul 3 4 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial -429 21197 a @beginspecial @setspecial 0.24 0.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 1.12 1.48 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 1.84 1.48 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 5040 w @beginspecial @setspecial 2.72 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 3.44 3.08 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4157 w @beginspecial @setspecial 4.4 4.4 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3906 w @beginspecial @setspecial 5.28 5.64 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 6 5.64 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 5039 w @beginspecial @setspecial 6.88 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 7.6 7.24 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4158 w @beginspecial @setspecial 8.56 8.56 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 9.44 9.8 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 10.16 9.8 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4158 w @beginspecial @setspecial 11.12 11.12 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3905 w @beginspecial @setspecial 12 12.36 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 12.72 12.36 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4914 w @beginspecial @setspecial 13.6 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 14.32 13.92 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 4283 w @beginspecial @setspecial 15.28 15.28 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 3906 w @beginspecial @setspecial 16.04 16.52 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial @beginspecial @setspecial 17.00 16.52 sub 1864679 mul 65536 div 72.27 div 72 mul 2 3 sub 1864679 mul 65536 div 72.27 div 72 mul .02 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ ] psline @endspecial 19661 -850 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3149 x @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3150 x @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6547 8598 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 26229 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6547 11748 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 18221 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 16016 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 1524 14898 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 10047 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 13197 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 11023 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 9985 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 1524 18047 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6992 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6110 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6992 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6110 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 8063 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6867 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 6236 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial -429 21197 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3905 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 5040 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4157 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3906 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 5039 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4158 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3905 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4158 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3905 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4914 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 4283 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3906 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial -429 24346 a @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2267 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3023 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2771 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3024 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2771 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3024 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2771 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2268 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2772 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2267 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3024 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 2394 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 3023 w @beginspecial @setspecial .2 1864679 mul 65536 div 72.27 div 72 mul [ 0 0 0 ] [ 1 1 1 ] .02 1864679 mul 65536 div 72.27 div 72 mul [ ] true pscircle @endspecial 19661 724 a @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 19336 1152 a Ft(2)19661 3874 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 19336 4302 a(1)13104 7024 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 12779 7452 a(1)26219 7024 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 25893 7452 a(3)6547 10173 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 6222 10601 a(2)28772 10173 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 28447 10601 a(2)36780 10173 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 36454 10601 a(4)4036 13323 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 3710 13751 a(3)9059 13323 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 8734 13751 a(3)24768 13323 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 24443 13751 a(3)38287 13323 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 37962 13751 a(3)43280 13323 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 42954 13751 a(7)1524 16472 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 1199 16900 a(3)10043 16472 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 9718 16900 a(3)13099 16472 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 12773 16900 a(5)23193 16472 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 22868 16900 a(3)26248 16472 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 25923 16900 a(5)35791 16472 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 35466 16900 a(5)44217 16472 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 43891 16900 a(5)47335 16472 y @beginspecial @setspecial 956519 65536 div 693290 65536 div psdeletebox @endspecial 46684 16900 a(11)547 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 222 20050 a(4)2500 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 2175 20050 a(6)8516 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 8191 20050 a(2)13650 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 13325 20050 a(6)15602 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 15277 20050 a(8)21618 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 21293 20050 a(6)26752 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 26427 20050 a(2)28705 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 28380 20050 a(8)34815 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 34490 20050 a(6)36768 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 36443 20050 a(8)42658 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 42332 20050 a(8)47917 19622 y @beginspecial @setspecial 571499 65536 div 693290 65536 div psdeletebox @endspecial 47592 20050 a(8)49870 19622 y @beginspecial @setspecial 956519 65536 div 693290 65536 div psdeletebox @endspecial 49220 20050 a(18)-429 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial -706 23128 a Fj(5)2910 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 2633 23128 a(5)4043 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 3767 23128 a(9)7949 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 7672 23128 a(5)9083 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 8806 23128 a(5)12673 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 12397 23128 a(5)16012 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 15735 23128 a(9)17146 22772 y @beginspecial @setspecial 841841 65536 div 608822 65536 div psdeletebox @endspecial 16592 23128 a(13)21051 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 20775 23128 a(5)22185 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 21908 23128 a(9)25776 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 25499 23128 a(5)29114 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 28837 23128 a(5)30248 22772 y @beginspecial @setspecial 841841 65536 div 608822 65536 div psdeletebox @endspecial 29695 23128 a(13)33839 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 33562 23128 a(9)37177 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 36900 23128 a(5)38311 22772 y @beginspecial @setspecial 841841 65536 div 608822 65536 div psdeletebox @endspecial 37758 23128 a(13)42154 22772 y @beginspecial @setspecial 514160 65536 div 608822 65536 div psdeletebox @endspecial 41877 23128 a(9)43287 22772 y @beginspecial @setspecial 841841 65536 div 608822 65536 div psdeletebox @endspecial 42734 23128 a(13)46941 22772 y @beginspecial @setspecial 841841 65536 div 608822 65536 div psdeletebox @endspecial 46387 23128 a(13)50091 22772 y @beginspecial @setspecial 841841 65536 div 608822 65536 div psdeletebox @endspecial 49537 23128 a(13)51602 22772 y @beginspecial @setspecial 841841 65536 div 608822 65536 div psdeletebox @endspecial 51049 23128 a(29)800 27496 y SDict begin H.S end 800 27496 a 800 27496 a SDict begin 14.5 H.A end 800 27496 a 800 27496 a SDict begin [ /View [/XYZ H.V] /Dest (figure.7) cvn H.B /DEST pdfmark end 800 27496 a Black 19652 30208 a Ft(Figure)433 b(7:)p 0 TeXcolorgray 25342 28603 a SDict begin H.S end 25342 28603 a 0 TeXcolorgray 0 TeXcolorgray 25342 28603 a SDict begin H.R end 25342 28603 a 25342 28603 a SDict begin [ /View [/XYZ H.V] /Dest (figure.7) cvn H.B /DEST pdfmark end 25342 28603 a Black 1605 x Ft(The)h(traces)f (tree)p Black Black 2751 33591 a(If)423 b(w)-36 b(e)423 b(denote)f(b)-36 b(y)423 b Fr(H)12998 33790 y Fy(n)14047 33591 y Ft(the)f(sum)g(of)h(the)f(v)-72 b(alues)424 b(of)f(the)f Fr(n)p Ft(-th)g(ro)-36 b(w)423 b(of)h(this)e(latter)h(tree,)i(w)-36 b(e)423 b(ha)-36 b(v)g(e)423 b(the)800 35196 y(follo)-36 b(wing)436 b(result:)800 35454 y SDict begin H.S end 800 35454 a 800 35454 a SDict begin 14.5 H.A end 800 35454 a 800 35454 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.15) cvn H.B /DEST pdfmark end 800 35454 a Black 2408 x Fo(Prop)42 b(osition)582 b(15.)p Black 591 w Fs(We)530 b(have)g Fr(H)18014 38061 y Fv(1)19031 37862 y Ft(=)490 b(2)p Fs(,)547 b Fr(H)23207 38061 y Fv(2)24224 37862 y Ft(=)490 b(1)p Fs(,)547 b Fr(H)28400 38061 y Fv(3)29417 37862 y Ft(=)490 b(4)531 b Fs(and,)547 b(for)531 b(any)f Fr(n)491 b Fq(\270)g Ft(3)p Fs(,)548 b Fr(H)45269 38061 y Fy(n)46385 37862 y Ft(=)491 b(2)p Fr(H)49616 38061 y Fy(n)p Fw(\241)p Fv(1)51788 37862 y Ft(+)800 39467 y Fr(H)1878 39666 y Fy(n)p Fw(\241)p Fv(3)4001 39467 y Fq(\241)296 b Ft(\()p Fq(\241)p Ft(1\))8025 38985 y Fy(n)8946 39467 y Fq(\242)g Ft(2)p Fs(.)p Black 800 42134 a(Pr)-66 b(o)g(of.)p Black 649 w Ft(W)-108 b(e)434 b(\257rst)f(pro)-36 b(v)g(e)433 b(the)g(follo)-36 b(wing)436 b(statemen)-36 b(t.)800 42392 y SDict begin H.S end 800 42392 a 800 42392 a SDict begin 14.5 H.A end 800 42392 a 800 42392 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.16) cvn H.B /DEST pdfmark end 800 42392 a Black 2011 x Fo(Prop)42 b(osition)500 b(16.)p Black 554 w Fs(With)464 b(the)g(same)h(notation)f(as)h(in)f(the)h(pr) -66 b(o)g(of)464 b(of)h(Pr)-66 b(op)g(osition)p 0 .5 0 TeXcolorrgb 42374 44403 a SDict begin H.S end 42374 44403 a 0 .5 0 TeXcolorrgb Fs(5)p 0 .5 0 TeXcolorrgb 43039 43542 a SDict begin H.R end 43039 43542 a 43039 44403 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.5) cvn H.B /ANN pdfmark end 43039 44403 a Black Fs(,)464 b(we)h(have)9638 47283 y Fr(G)10664 46735 y Fw(\241)10664 47612 y Fv(3)11820 47283 y Ft(=)369 b Fr(G)14227 46735 y Fw(\241)14227 47612 y Fv(4)15383 47283 y Ft(=)g(1)1328 b Fr(G)19768 46735 y Fw(\241)19768 47612 y Fy(n)20924 47283 y Ft(=)369 b(2)p Fr(G)23981 46735 y Fw(\241)23981 47612 y Fy(n)p Fw(\241)p Fv(1)26105 47283 y Ft(+)295 b Fr(G)28438 46735 y Fw(\241)28438 47612 y Fy(n)p Fw(\241)p Fv(3)30562 47283 y Ft(+)f(\()p Fq(\241)p Ft(1\))34563 46735 y Fy(n)35485 47283 y Fq(\242)h Ft(2)465 b Fs(\(for)g Fr(n)370 b Fq(\270)f Ft(5)p Fs(\))q Fr(;)8638 50164 y(G)9664 49615 y Fv(+)9664 50492 y(3)10820 50164 y Ft(=)g(3)1329 b Fr(G)15206 49615 y Fv(+)15206 50492 y(4)16362 50164 y Ft(=)368 b(8)1329 b Fr(G)20747 49615 y Fv(+)20747 50492 y Fy(n)21903 50164 y Ft(=)369 b(2)p Fr(G)24960 49615 y Fv(+)24960 50492 y Fy(n)p Fw(\241)p Fv(1)27084 50164 y Ft(+)295 b Fr(G)29417 49615 y Fv(+)29417 50492 y Fy(n)p Fw(\241)p Fv(3)31540 50164 y Fq(\241)h Ft(\()p Fq(\241)p Ft(1\))35564 49615 y Fy(n)36485 50164 y Fq(\242)f Ft(2)466 b Fs(\(for)e Fr(n)370 b Fq(\270)f Ft(5)p Fs(\))q Fr(:)p Black 800 52750 a Fs(Pr)-66 b(o)g(of.)p Black 649 w Ft(The)487 b(ab)36 b(o)-36 b(v)g(e)487 b(relations)h(are)f (routinely)g(v)-36 b(eri\257ed)486 b(for)i(the)e(\257rst)g(v)-72 b(alues)487 b(of)h Fr(n)p Ft(.)738 b(Let)487 b(assume)f(that)800 54355 y(these)422 b(relations)i(are)f(true)f(for)h(all)h(indices)f(un) -36 b(til)422 b Fr(n)273 b Fq(\241)h Ft(1.)575 b(Recalling)424 b(the)e(relation)i Fr(G)42975 53873 y Fw(\241)42975 54683 y Fy(n)44131 54355 y Ft(=)368 b Fr(G)46537 53805 y Fv(+)46537 54681 y Fy(n)p Fw(\241)p Fv(1)48639 54355 y Fq(\241)274 b Fr(G)50972 54554 y Fy(n)p Fw(\241)p Fv(2)800 55960 y Ft(\(see)434 b(the)f(pro)36 b(of)434 b(of)g(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 17524 55960 a SDict begin H.S end 17524 55960 a 0 .5 0 TeXcolorrgb Ft(5)p 0 .5 0 TeXcolorrgb 18175 55104 a SDict begin H.R end 18175 55104 a 18175 55960 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.5) cvn H.B /ANN pdfmark end 18175 55960 a Black Ft(\))433 b(w)-36 b(e)434 b(get)10737 58840 y Fr(G)11763 58292 y Fw(\241)11763 59169 y Fy(n)13657 58840 y Ft(=)1106 b Fr(G)16801 58292 y Fv(+)16801 59169 y Fy(n)p Fw(\241)p Fv(1)18925 58840 y Fq(\241)295 b Fr(G)21279 59039 y Fy(n)p Fw(\241)p Fv(2)13657 60777 y Ft(=)15775 59701 y Fm(\241)16384 60777 y Ft(2)p Fr(G)18060 60229 y Fv(+)18060 61106 y Fy(n)p Fw(\241)p Fv(2)20184 60777 y Ft(+)g Fr(G)22517 60229 y Fv(+)22517 61106 y Fy(n)p Fw(\241)p Fv(4)24641 60777 y Fq(\241)g Ft(\()p Fq(\241)p Ft(1\))28664 60229 y Fy(n)p Fw(\241)p Fv(1)30788 60777 y Fq(\242)g Ft(2)32102 59701 y Fm(\242)33006 60777 y Fq(\241)34335 59701 y Fm(\241)34944 60777 y Fr(G)35970 60229 y Fv(+)35970 61106 y Fy(n)p Fw(\241)p Fv(2)38093 60777 y Ft(+)g Fr(G)40426 60229 y Fw(\241)40426 61106 y Fy(n)p Fw(\241)p Fv(2)42255 59701 y Fm(\242)13657 62715 y Ft(=)1106 b Fr(G)16801 62166 y Fv(+)16801 63043 y Fy(n)p Fw(\241)p Fv(2)18925 62715 y Fq(\241)295 b Fr(G)21279 62166 y Fw(\241)21279 63043 y Fy(n)p Fw(\241)p Fv(2)23403 62715 y Ft(+)g Fr(G)25736 62166 y Fv(+)25736 63043 y Fy(n)p Fw(\241)p Fv(4)27860 62715 y Ft(+)f(\()p Fq(\241)p Ft(1\))31861 62166 y Fy(n)32783 62715 y Fq(\242)h Ft(2)p Fr(:)2751 65595 y Ft(W)-108 b(e)531 b(th)-36 b(us)531 b(ha)-36 b(v)g(e)531 b(to)g(v)-36 b(erify)533 b(the)e(relation)h Fr(G)24773 65045 y Fv(+)24773 65921 y Fy(n)p Fw(\241)p Fv(2)26963 65595 y Fq(\241)362 b Fr(G)29384 65045 y Fw(\241)29384 65921 y Fy(n)p Fw(\241)p Fv(2)31574 65595 y Ft(+)f Fr(G)33973 65045 y Fv(+)33973 65921 y Fy(n)p Fw(\241)p Fv(4)36337 65595 y Ft(=)535 b(2)p Fr(G)39560 65045 y Fw(\241)39560 65921 y Fy(n)p Fw(\241)p Fv(1)41751 65595 y Ft(+)361 b Fr(G)44150 65045 y Fw(\241)44150 65921 y Fy(n)p Fw(\241)p Fv(3)45979 65595 y Ft(.)871 b(W)-108 b(e)531 b(write,)800 67200 y(using)433 b(again)i(the)e(relation)h Fr(G)15707 66718 y Fw(\241)15707 67528 y Fy(n)16863 67200 y Ft(=)369 b Fr(G)19270 66650 y Fv(+)19270 67526 y Fy(n)p Fw(\241)p Fv(1)21393 67200 y Fq(\241)296 b Fr(G)23748 67399 y Fy(n)p Fw(\241)p Fv(2)25576 67200 y Ft(:)p Black 26150 70521 a(25)p Black eop %%Page: 26 26 26 25 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.26) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 5041 262 a Ft(2)p Fr(G)6717 -287 y Fw(\241)6717 590 y Fy(n)p Fw(\241)p Fv(1)8841 262 y Ft(+)295 b Fr(G)11174 -287 y Fw(\241)11174 590 y Fy(n)p Fw(\241)p Fv(3)14109 262 y Ft(=)1107 b(2\()p Fr(G)18410 -287 y Fv(+)18410 590 y Fy(n)p Fw(\241)p Fv(2)20533 262 y Fq(\241)296 b Fr(G)22888 461 y Fy(n)p Fw(\241)p Fv(3)24716 262 y Ft(\))f(+)g(\()p Fr(G)28356 -287 y Fv(+)28356 590 y Fy(n)p Fw(\241)p Fv(4)30480 262 y Fq(\241)g Fr(G)32834 461 y Fy(n)p Fw(\241)p Fv(5)34663 262 y Ft(\))14109 2199 y(=)1107 b(\()p Fr(G)17760 1651 y Fv(+)17760 2527 y Fy(n)p Fw(\241)p Fv(2)19883 2199 y Fq(\241)295 b Fr(G)22237 1651 y Fw(\241)22237 2527 y Fy(n)p Fw(\241)p Fv(2)24361 2199 y Ft(+)g Fr(G)26694 1651 y Fv(+)26694 2527 y Fy(n)p Fw(\241)p Fv(4)28522 2199 y Ft(\))g(+)g(\()p Fr(G)32162 1651 y Fv(+)32162 2527 y Fy(n)p Fw(\241)p Fv(2)34286 2199 y Ft(+)g Fr(G)36619 1651 y Fw(\241)36619 2527 y Fy(n)p Fw(\241)p Fv(2)38742 2199 y Fq(\241)h Ft(2)p Fr(G)41747 2398 y Fy(n)p Fw(\241)p Fv(3)43871 2199 y Fq(\241)f Fr(G)46225 2398 y Fy(n)p Fw(\241)p Fv(5)48054 2199 y Ft(\))14109 4136 y(=)1107 b(\()p Fr(G)17760 3588 y Fv(+)17760 4465 y Fy(n)p Fw(\241)p Fv(2)19883 4136 y Fq(\241)295 b Fr(G)22237 3588 y Fw(\241)22237 4465 y Fy(n)p Fw(\241)p Fv(2)24361 4136 y Ft(+)g Fr(G)26694 3588 y Fv(+)26694 4465 y Fy(n)p Fw(\241)p Fv(4)28522 4136 y Ft(\))g(+)g(\()p Fr(G)32162 4335 y Fy(n)p Fw(\241)p Fv(2)34286 4136 y Fq(\241)g Fr(G)36640 4335 y Fy(n)p Fw(\241)p Fv(2)38469 4136 y Ft(\))p Fr(;)800 7070 y Ft(so)434 b(the)f(desired)g(relation)h(for)g Fr(G)16834 6588 y Fw(\241)16834 7398 y Fy(n)18055 7070 y Ft(is)g(v)-36 b(eri\257ed.)2751 8675 y(W)-108 b(e)434 b(can)f(then)g(write)12821 11608 y Fr(G)13847 11060 y Fv(+)13847 11937 y Fy(n)15741 11608 y Ft(=)1106 b Fr(G)18885 11807 y Fy(n)19807 11608 y Fq(\241)295 b Fr(G)22161 11060 y Fw(\241)22161 11937 y Fy(n)15741 13545 y Ft(=)1106 b Fr(G)18885 13744 y Fy(n)19807 13545 y Fq(\241)21135 12469 y Fm(\241)21744 13545 y Ft(2)p Fr(G)23420 12997 y Fw(\241)23420 13874 y Fy(n)p Fw(\241)p Fv(1)25544 13545 y Ft(+)295 b Fr(G)27877 12997 y Fw(\241)27877 13874 y Fy(n)p Fw(\241)p Fv(3)30000 13545 y Ft(+)g(\()p Fq(\241)p Ft(1\))34002 12997 y Fy(n)34924 13545 y Fq(\242)g Ft(2\))15741 15482 y(=)1106 b Fr(G)18885 15681 y Fy(n)19807 15482 y Fq(\241)295 b Ft(2)p Fr(G)22811 14934 y Fw(\241)22811 15811 y Fy(n)p Fw(\241)p Fv(1)24935 15482 y Fq(\241)g Fr(G)27289 14934 y Fw(\241)27289 15811 y Fy(n)p Fw(\241)p Fv(3)29413 15482 y Fq(\241)h Ft(\()p Fq(\241)p Ft(1\))33437 14934 y Fy(n)34358 15482 y Fq(\242)f Ft(2)15741 17420 y(=)1106 b Fr(G)18885 17619 y Fy(n)19807 17420 y Fq(\241)295 b Ft(2)p Fr(G)22811 17619 y Fy(n)p Fw(\241)p Fv(1)24935 17420 y Ft(+)g(2)p Fr(G)27918 16871 y Fv(+)27918 17748 y Fy(n)p Fw(\241)p Fv(1)30042 17420 y Fq(\241)g Fr(G)32396 16871 y Fw(\241)32396 17748 y Fy(n)p Fw(\241)p Fv(3)34520 17420 y Fq(\241)g Ft(\()p Fq(\241)p Ft(1\))38543 16871 y Fy(n)39465 17420 y Fq(\242)g Ft(2)15741 19357 y(=)1106 b Fr(G)18885 19556 y Fy(n)p Fw(\241)p Fv(3)21009 19357 y Ft(+)295 b(2)p Fr(G)23992 18808 y Fv(+)23992 19685 y Fy(n)p Fw(\241)p Fv(1)26116 19357 y Fq(\241)g Fr(G)28470 18808 y Fw(\241)28470 19685 y Fy(n)p Fw(\241)p Fv(3)30594 19357 y Fq(\241)g Ft(\()p Fq(\241)p Ft(1\))34617 18808 y Fy(n)35539 19357 y Fq(\242)g Ft(2)15741 21294 y(=)1106 b(2)p Fr(G)19535 20746 y Fv(+)19535 21622 y Fy(n)p Fw(\241)p Fv(1)21659 21294 y Ft(+)295 b Fr(G)23992 20746 y Fv(+)23992 21622 y Fy(n)p Fw(\241)p Fv(3)26116 21294 y Fq(\241)g Ft(\()p Fq(\241)p Ft(1\))30139 20746 y Fy(n)31061 21294 y Fq(\242)g Ft(2)p Fr(;)800 24227 y Ft(so)434 b(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 9395 24227 a SDict begin H.S end 9395 24227 a 0 .5 0 TeXcolorrgb Ft(16)p 0 .5 0 TeXcolorrgb 10696 23371 a SDict begin H.R end 10696 23371 a 10696 24227 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.16) cvn H.B /ANN pdfmark end 10696 24227 a Black 433 w Ft(is)434 b(pro)-36 b(v)g(ed.)p 51860 25833 45 878 v 51905 24999 781 45 v 51905 25833 V 52684 25833 45 878 v 2751 28434 a(W)-108 b(e)323 b(no)-36 b(w)322 b(come)h(bac)-36 b(k)323 b(to)f(the)g(pro)36 b(of)323 b(of)g(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 28764 28434 a SDict begin H.S end 28764 28434 a 0 .5 0 TeXcolorrgb Ft(15)p 0 .5 0 TeXcolorrgb 30065 27578 a SDict begin H.R end 30065 27578 a 30065 28434 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.15) cvn H.B /ANN pdfmark end 30065 28434 a Black Ft(.)541 b(Recall)324 b(\(see)f(commen)-36 b(ts)322 b(after)h(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 800 30039 a SDict begin H.S end 800 30039 a 0 .5 0 TeXcolorrgb Ft(5)p 0 .5 0 TeXcolorrgb 1450 29183 a SDict begin H.R end 1450 29183 a 1450 30039 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.5) cvn H.B /ANN pdfmark end 1450 30039 a Black Ft(\))409 b(that)g(if)h(w)-36 b(e)410 b(write)f Fr(v)12146 30238 y Fy(n)13141 30039 y Ft(=)369 b Fr(\256)15349 30238 y Fy(n)15975 30039 y Fr(a)245 b Ft(+)g Fr(\257)18894 30238 y Fy(n)19521 30039 y Fr(b)409 b Ft(for)h(the)f(sum)f(of)i(the)f Fr(n)p Ft(-th)g(ro)-36 b(w)409 b(of)h(the)f(general)g(restricted)g (tree,)800 31644 y(then)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)8319 34578 y Fr(\256)9146 34777 y Fv(0)10041 34578 y Ft(=)369 b(1)1301 b Fr(\256)14200 34777 y Fv(1)15095 34578 y Ft(=)369 b(0)1301 b Fr(\256)19254 34777 y Fv(2)20149 34578 y Ft(=)368 b(1)1301 b Fr(\256)24307 34777 y Fv(3)25202 34578 y Ft(=)369 b(2)1301 b Fr(\256)29361 34777 y Fy(n)30356 34578 y Ft(=)369 b(2)p Fr(\256)33214 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Ft(exactly)i(the)e(sum)g(of)i(the)e(\257rst)f (co)36 b(e\261cien)-36 b(ts)525 b(of)g(these)f(matrices,)800 43046 y(since)496 b(some)h Fr(a)p Ft(-co)36 b(e\261cien)-36 b(ts)496 b(in)g(the)g(\()p Fr(n)338 b Fq(\241)g Ft(1\)-th)496 b(ro)-36 b(w)496 b(ha)-36 b(v)g(e)497 b(to)f(b)36 b(e)496 b(coun)-36 b(ted)496 b(t)-36 b(wice.)767 b(W)-108 b(e)496 b(can)g(consider)800 44651 y Fr(\256)1627 44850 y Fy(n)p Fw(\241)p Fv(1)3899 44651 y Ft(as)443 b(the)f(sum)g(of)i(the)e(\257rst) g(co)36 b(e\261cien)-36 b(ts)443 b(of)g(all)h(matrices)f(whic)-36 b(h)443 b(are)f(righ)-36 b(t)443 b(c)-36 b(hildren.)605 b(It)443 b(remains,)800 46256 y(then,)502 b(to)489 b(ev)-72 b(aluate)489 b(the)f(sum)g(of)i Fr(a)p Ft(-co)36 b(e\261cien)-36 b(ts)488 b(of)h(elemen)-36 b(ts)489 b(of)g(the)f(\()p Fr(n)333 b Fq(\241)g Ft(1\)-th)488 b(ro)-36 b(w)489 b(whic)-36 b(h)489 b(are)f(left)800 47861 y(c)-36 b(hildren.)547 b(F)-108 b(or)340 b(this,)359 b(let)341 b(us)g(consider)f(the)h (particular)g(case)g Fr(a)369 b Ft(=)f(1)341 b(and)g Fr(b)369 b Ft(=)f(0)341 b(in)g(the)f(general)i(restricted)800 49466 y(tree.)568 b(Let)404 b(us)g(denote)g(b)-36 b(y)404 b Fr(\256)14612 48984 y Fw(\241)14604 49795 y Fy(n)15805 49466 y Ft(the)f(sum)h(of)h(left)g(c)-36 b(hildren)404 b(in)g(the)g Fr(n)p Ft(-th)f(ro)-36 b(w)405 b(of)g(this)f(tree:)564 b(w)-36 b(e)404 b(then)g(ha)-36 b(v)g(e)800 51071 y Fr(H)1878 51270 y Fy(n)2873 51071 y Ft(=)369 b Fr(\256)5081 51270 y Fy(n)p Fw(\241)p Fv(1)6959 51071 y Ft(+)49 b Fr(\256)8855 50521 y Fw(\241)8847 51397 y Fy(n)p Fw(\241)p Fv(1)10724 51071 y Ft(+)g Fr(\257)12519 51270 y Fy(n)13146 51071 y Ft(.)538 b(W)-108 b(e)313 b(kno)-36 b(w)314 b(b)-36 b(y)313 b(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 28003 51071 a SDict begin H.S end 28003 51071 a 0 .5 0 TeXcolorrgb Ft(16)p 0 .5 0 TeXcolorrgb 29304 50215 a SDict begin H.R end 29304 50215 a 29304 51071 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.16) cvn H.B /ANN pdfmark end 29304 51071 a Black 313 w Ft(and)313 b(b)-36 b(y)313 b(the)f(fact)i(men)-36 b(tioned)312 b(in)h(the)g(b)36 b(egining)800 52676 y(of)479 b(the)e(presen)-36 b(t)477 b(section)h(that,)489 b(for)479 b(all)f Fr(n)445 b Fq(\270)g Ft(3,)490 b(w)-36 b(e)478 b(ha)-36 b(v)g(e)478 b Fr(\256)30719 52127 y Fw(\241)30711 53002 y Fy(n)p Fw(\241)p Fv(1)32985 52676 y Ft(=)444 b(2)p Fr(\256)35926 52127 y Fw(\241)35918 53002 y Fy(n)p Fw(\241)p Fv(2)38072 52676 y Ft(+)325 b Fr(\256)40244 52127 y Fw(\241)40236 53002 y Fy(n)p Fw(\241)p Fv(4)42391 52676 y Ft(+)g(\()p Fq(\241)p Ft(1\))46423 52194 y Fy(n)p Fw(\241)p Fv(1)48577 52676 y Fq(\242)g Ft(2.)712 b(W)-108 b(e)800 54281 y(th)-36 b(us)515 b(obtain)h(the)f (righ)-36 b(t)515 b(recurrence)g(relation)i(for)f(the)f(sequence)h(\()p Fr(H)36089 54480 y Fy(n)36715 54281 y Ft(\))37221 54480 y Fy(n)37847 54281 y Ft(,)537 b(and)515 b(a)h(simple)g(calculation)800 55887 y(giv)-36 b(es)435 b(the)e(\257rst)g(v)-72 b(alues)434 b(of)g(this)f(sequence.)p 51860 55887 45 878 v 51905 55053 781 45 v 51905 55887 V 52684 55887 45 878 v 2751 58488 a(The)512 b(follo)-36 b(wing)515 b(sho)-36 b(ws)512 b(that)g(the)f(structure)g(of)i(the)f(trace)g(tree)g(is)g(more)h (di\261cult)e(to)i(apprehend)800 60093 y(than)433 b(these)g(of)h(the)f (other)h(trees)f(considered)g(here.)800 60093 y SDict begin H.S end 800 60093 a 800 60093 a SDict begin 14.5 H.A end 800 60093 a 800 60093 a SDict begin [ /View [/XYZ H.V] /Dest (Lemme.17) cvn H.B /DEST pdfmark end 800 60093 a Black 2712 x Fo(Lemma)451 b(17.)p Black 521 w Fs(L)-66 b(et)426 b(us)h(denote)e(by)h Fr(t)18574 63004 y Fy(n)19626 62805 y Fs(the)g(lab)-66 b(el)426 b(of)g(the)g Fr(n)p Fs(-th)h(e)-66 b(dge)425 b(given)g(by)h(a)g(walk)h Ft(\()p Fr(w)44234 63004 y Fy(n)44860 62805 y Ft(\))45366 63004 y Fy(n)46418 62805 y Fs(in)f(the)g(tr)-66 b(ac)g(e)800 64410 y(tr)g(e)g(e.)p Black 2737 67122 a Fq(\262)p Black 651 w Fs(If)464 b Fr(w)6346 67321 y Fy(n)7341 67122 y Ft(=)368 b Fr(w)9651 67321 y Fy(n)p Fv(+1)11945 67122 y Fs(\(thus)466 b(a)f Ft(+)f 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Fm(\265)11109 36305 y Fr(F)11951 36504 y Fy(k)14228 36305 y Fr(L)15113 36504 y Fy(k)24 b Fv(+1)10508 37910 y Fr(F)11350 38109 y Fy(k)g Fw(\241)p Fv(2)14228 37910 y Fr(L)15113 38109 y Fy(k)g Fw(\241)p Fv(1)17438 35246 y Fm(\266)18889 37119 y Ft(for)474 b(the)f(\257rst)f(w)-36 b(alk)475 b(and)31560 35246 y Fm(\265)33091 36305 y Fr(F)33933 36504 y Fy(k)24 b Fv(+3)36812 36305 y Fr(F)37654 36504 y Fy(k)g Fv(+1)33091 37910 y Fr(F)33933 38109 y Fy(k)g Fv(+1)36812 37910 y Fr(F)37654 38109 y Fy(k)g Fw(\241)p Fv(1)39978 35246 y Fm(\266)41429 37119 y Ft(for)474 b(the)f(second.)698 b(The)800 39425 y(traces)434 b(are)g Fr(F)7575 39624 y Fy(k)8439 39425 y Ft(+)294 b Fr(L)10630 39624 y Fy(k)24 b Fw(\241)p Fv(1)12835 39425 y Ft(and)433 b Fr(F)16206 39624 y Fy(k)24 b Fw(\241)p Fv(1)18272 39425 y Ft(+)295 b Fr(F)20421 39624 y Fy(k)24 b Fv(+3)22192 39425 y Ft(,)434 b(whic)-36 b(h)434 b(are)f(di\256eren)-36 b(t)433 b(n)-36 b(um)g(b)36 b(ers)432 b(for)i(an)-36 b(y)434 b Fr(k)45 b Ft(.)p 51860 39425 45 878 v 51905 38591 781 45 v 51905 39425 V 52684 39425 45 878 v 800 40567 a SDict begin H.S end 800 40567 a 800 40567 a SDict begin 14.5 H.A end 800 40567 a 800 40567 a SDict begin [ /View [/XYZ H.V] /Dest (Cor.6) cvn H.B /DEST pdfmark end 800 40567 a Black 1133 x Fo(Corollary)521 b(6.)p Black 554 w Fs(The)464 b(pr)-66 b(evious)465 b(function)e Ft(\()p Fr(w)23479 41899 y Fy(n)24105 41700 y Ft(\))24611 41899 y Fy(n)25606 41700 y Fq(\241)-221 b(!)369 b Ft(\()p Fr(t)29091 41899 y Fy(n)29717 41700 y Ft(\))30223 41899 y Fy(n)31314 41700 y Fs(is)465 b(not)f(of)h(\257nite)e(typ)-66 b(e.)800 43815 y SDict begin H.S end 800 43815 a 800 43815 a SDict begin 14.5 H.A end 800 43815 a 800 43815 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.4) cvn H.B /DEST pdfmark end 800 43815 a 1661 x Fn(4.4)1793 b(Size)598 b(of)f(w)-50 b(alks)599 b(in)f(R)800 46334 y SDict begin H.S end 800 46334 a 800 46334 a SDict begin 14.5 H.A end 800 46334 a 800 46334 a SDict begin [ /View [/XYZ H.V] /Dest (Notation.4) cvn H.B /DEST pdfmark end 800 46334 a Black 1605 x Fo(Notation)521 b(4.)p Black 563 w Fs(F)-100 b(or)482 b Fr(a)g Fs(and)f Fr(b)h Fs(r)-66 b(elatively)480 b(prime,)k(we)e(denote)f(by)g Fr(\275)p Ft(\()p Fr(a;)221 b(b)p Ft(\))482 b Fs(the)f(smal)66 b(lest)482 b(inte)-66 b(ger)479 b Fr(n)j Fs(such)800 49544 y(that)344 b(the)g Fr(n)p Fs(-th)h(r)-66 b(ow)345 b(of)f(e)-66 b(dges)344 b(in)f Fo(T)i Fs(c)-66 b(ontains)343 b(an)h(e)-66 b(dge)344 b(with)g Fr(a)g Fs(as)h(p)-66 b(ar)g(ent)344 b(and)g Fr(b)g Fs(as)h(child.)557 b(As)345 b(establishe)-66 b(d)800 51149 y(in)497 b(Pr)-66 b(op)g(osition)p 0 .5 0 TeXcolorrgb 9368 51149 a SDict begin H.S end 9368 51149 a 0 .5 0 TeXcolorrgb Fs(3)p 0 .5 0 TeXcolorrgb 10032 50288 a SDict begin H.R end 10032 50288 a 10032 51149 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.3) cvn H.B /ANN pdfmark end 10032 51149 a Black 498 w Fs(and)497 b(Pr)-66 b(op)g(osition)p 0 .5 0 TeXcolorrgb 20027 51149 a SDict begin H.S end 20027 51149 a 0 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b(elatively)540 b(prime)g(inte)-66 b(gers)539 b Fr(a)800 54359 y Fs(and)450 b Fr(b)h Fs(b)-66 b(eing)448 b(given,)j(any)f(p)-66 b(air)450 b(of)g(the)g(form)f Ft(\()p Fr(r)24400 54558 y Fy(i)24776 54359 y Fr(;)221 b(r)25944 54558 y Fy(i)p Fv(+1)27523 54359 y Ft(\))450 b Fs(\(r)-66 b(esp.)592 b Ft(\()p Fr(r)33368 54558 y Fy(i)p Fv(+1)34946 54359 y Fr(;)221 b(r)36114 54558 y Fy(i)36491 54359 y Ft(\))p Fs(\))450 b(is)g(said)h(in)e Ft(ascending)h Fs(\(r)-66 b(esp.)800 55964 y Ft(descending)p Fs(\))500 b Ft(order)p Fs(.)705 b(When)500 b(after)g Ft(\()p Fr(r)20072 56163 y Fy(i)20448 55964 y Fr(;)221 b(r)21616 56163 y Fy(i)p Fv(+1)23195 55964 y Ft(\))500 b Fs(c)-66 b(omes)500 b Ft(\()p Fr(r)29181 56163 y Fy(i)29557 55964 y Fr(;)221 b(r)30725 56163 y Fy(i)p Fw(\241)p Fv(1)32304 55964 y Ft(\))p Fs(,)509 b(or)501 b(after)f Ft(\()p Fr(r)39664 56163 y Fy(i)p Fv(+1)41242 55964 y Fr(;)221 b(r)42410 56163 y Fy(i)42786 55964 y Ft(\))501 b Fs(c)-66 b(omes)500 b Ft(\()p Fr(r)48773 56163 y Fy(i)p Fw(\241)p Fv(1)50351 55964 y Fr(;)221 b(r)51519 56163 y Fy(i)51896 55964 y Ft(\))p Fs(,)800 57569 y(we)465 b(say)h(that)e(ther)-66 b(e)464 b(is)h(an)g Ft(in)-36 b(v)g(ersion)p Fs(.)2751 59174 y(If)487 b Ft([)p Fr(n)5275 59373 y Fv(0)5801 59174 y Fr(;)221 b(n)7159 59373 y Fv(1)7686 59174 y Fr(;)g(:)g(:)g(:)j(;)d(n) 11375 59373 y Fy(N)94 b Fw(\241)p Fv(1)13475 59174 y Ft(])487 b Fs(is)h(the)f(c)-66 b(ontinue)g(d)485 b(fr)-66 b(action)485 b(exp)-66 b(ansion)486 b(of)h Fr(a=b)p Fs(,)493 b(then)487 b(let)g(de\257ne)f Fr(\260)72 b Ft(\()p Fr(a;)221 b(b)p Ft(\))488 b Fs(as)800 60780 y(the)470 b(total)g(numb)-66 b(er)468 b(of)i(even)f(and)h(p)-66 b(ositive)468 b(p)-66 b(artial)470 b(quotients)f Fr(n)32772 60979 y Fy(i)33618 60780 y Fs(with)h Fr(i)378 b(<)g(N)437 b Fq(\241)299 b Ft(1)471 b Fs(\(note)e(that)h(the)g(last)800 62385 y(p)-66 b(artial)458 b(quotient,)g Fr(n)11079 62584 y Fy(N)94 b Fw(\241)p Fv(1)13178 62385 y Fs(,)460 b(do)-66 b(es)459 b(not)f(c)-66 b(ount,)458 b(neither)g Fr(n)28289 62584 y Fv(0)29274 62385 y Fs(in)f(the)i(c)-66 b(ase)458 b Fr(a)369 b(<)f(b)p Fs(\))459 b(and)g(by)f Fr(!)48 b Ft(\()p Fr(a;)221 b(b)p Ft(\))459 b Fs(the)f(total)800 63990 y(numb)-66 b(er)498 b(of)h(o)-66 b(dd)500 b(p)-66 b(artial)499 b(quotients)f Fr(n)19947 64189 y Fy(i)20822 63990 y Fs(with)i Fr(i)432 b(<)h(N)459 b Fq(\241)320 b Ft(1)p Fs(.)701 b(We)499 b(denote)g(by)g Fr(!)39772 63508 y Fw(\241)40559 63990 y Ft(\()p Fr(a;)221 b(b)p Ft(\))499 b Fs(\(r)-66 b(esp.)701 b Fr(!)48652 63508 y Fv(+)49439 63990 y Ft(\()p Fr(a;)221 b(b)p Ft(\))p Fs(\))800 65595 y(the)491 b(numb)-66 b(er)490 b(of)h(o)-66 b(dd)492 b(p)-66 b(artial)491 b(quotients)f Fr(n)22092 65794 y Fy(i)22960 65595 y Fs(with)h Fr(i)417 b(<)h(N)453 b Fq(\241)315 b Ft(1)492 b Fs(such)g(that)f(the)g(numb)-66 b(er)490 b(of)h(even)g(p)-66 b(artial)800 67200 y(quotients)464 b(among)g Fr(n)11306 67399 y Fy(i)p Fv(+1)12885 67200 y Fs(,)g Fr(n)14524 67399 y Fy(i)p Fv(+2)16103 67200 y Fs(,.)199 b(.)g(.)g(,)463 b Fr(n)19934 67399 y Fy(N)94 b Fw(\241)p Fv(2)22498 67200 y Fs(is)465 b(o)-66 b(dd)465 b(\(r)-66 b(esp.)597 b(even\).)p Black 26150 70521 a Ft(27)p Black eop %%Page: 28 28 28 27 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.28) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (Prop.17) cvn H.B /DEST pdfmark end 800 -4000 a Black 1328 x Fo(Prop)42 b(osition)440 b(17.)p Black 514 w Fs(L)-66 b(et)416 b Ft([)p Fr(n)14569 -2473 y Fv(0)15095 -2672 y Fr(;)221 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0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.1) cvn H.B /ANN pdfmark end 36445 36703 a Black 434 w Ft(and)f(Prop)36 b(osition)p 0 .5 0 TeXcolorrgb 46406 36703 a SDict begin H.S end 46406 36703 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 47056 35847 a SDict begin H.R end 47056 35847 a 47056 36703 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Prop.1) cvn H.B /ANN pdfmark end 47056 36703 a Black Ft(.)2751 38308 y(By)479 b(a)g(simple)g(application)g(of)h(Lemma)p 0 .5 0 TeXcolorrgb 23044 38308 a SDict begin H.S end 23044 38308 a 0 .5 0 TeXcolorrgb Ft(1)p 0 .5 0 TeXcolorrgb 23695 37452 a SDict begin H.R end 23695 37452 a 23695 38308 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Lemme.1) cvn H.B /ANN pdfmark end 23695 38308 a Black Ft(,)490 b(the)478 b(n)-36 b(um)g(b)36 b(er)477 b(of)j(steps)e(to)h(go)g(from)g(the)f(edge)h(\(1)p Fr(;)221 b Ft(1\))480 b(to)800 39913 y(the)504 b(edge)g(\(1)p Fr(;)221 b(n)8658 40112 y Fy(N)94 b 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