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\begin{center}
\vskip 1cm{\LARGE\bf 
Interspersions and Fractal Sequences \\
\vskip .1in
Associated with Fractions $c^j/d^k$
}
\vskip 1cm
\large
Clark Kimberling\\
Department of Mathematics\\
University of Evansville\\
1800 Lincoln Avenue\\
Evansville, IN 47722\\
USA\\
\href{mailto:ck6@evansville.edu}{\tt ck6@evansville.edu}
\end{center}

\vskip .2 in

\begin{abstract}
Suppose $c\geq 2$ and $d\geq 2$ are integers, and let $S$ be the set of
integers $\left\lfloor c^j/d^k\right\rfloor$, where $j$ and $k$ range
over the nonnegative integers.  Assume that $c$ and $d$ are multiplicatively
independent; that is,
if $p$ and $q$ are integers for which $c^p=d^q,$ then 
$p=q=0$.  The numbers in $S$ form interspersions in various ways.  Related
fractal sequences and permutations of the set of nonnegative integers are
also discussed.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}


\section{Introduction}

Throughout this article, the letters $c,d,j,k,p,q,h,m,n$ represent
nonnegative integers such that $c\geq 2$ and $d\geq 2,$ and $c$ and $d$ are
multiplicatively independent; that is,
if $c^{p}=d^{q},$ then $p=q=0.$ 

Definitions, examples, and references for the terms \textit{interspersion}
and \textit{fractal sequence} are easily accessible
(\cite{MathWorld1,MathWorld2,MathWorld3,SloCS}), so that only a brief summary
is given in this introduction.  This introduction also presents certain new
arrays defined from the manner in which the fractions $c^{j}/d^{k}$ are
distributed.  The main purpose of the article is to prove that each such
array is an interspersion.

\bigskip

\textbf{Definition.} An array $A=(a_{mh}),$ $m\geq 1,h\geq 1,$ of positive
integers is an \textit{interspersion} if

(I1)  the rows of $A$ partition the positive integers;

(I2)  every row of $A$ is an increasing sequence;

(I3)  every column of $A$ is an increasing (possibly finite) sequence;

(I4)  if $(u_{h})$ and $(v_{h})$ are distinct rows of $A,$ and $p$ and $q$
are indices for which $u_{p}<v_{q}<u_{p+1},$ then
\begin{equation*}
u_{p+1}<v_{q+1}<u_{p+2}.
\end{equation*}
Example 1 below illustrates the manner in which property (I4) matches the
name ``interspersion''; viz., the terms of
each row individually separate and are separated by the terms of all other
rows (after initial terms).

\bigskip

\textbf{Definition of the array} $T_{(c,d,k_{0})}=\{t(m,h)\}$.  Row 1 is
defined by $t(1,h)=c^{h-1},$ for $h=1,2,\ldots $.  For $m\geq 2,$ the first
term $t(m,1)$ of row $m$ is the least positive integer
\begin{equation*}
\left\lfloor c^{j}/d^{k}\right\rfloor ,\text{ \ \ where }k\geq k_{0},
\end{equation*}
that is not in rows $1,2,...,m-1.$  In order to define the rest of row $m,$
we shall choose a precise $k$ for the representation $t(m,1)=\left\lfloor
c^{j}/d^{k}\right\rfloor .$  According to Lemma~\ref{lemma2} below,
every $n$ has
infinitely many representations $\left\lfloor c^{j}/d^{k}\right\rfloor ,$
and we choose the one for which $k$ is minimal (with $k\geq k_{0})$, noting
that $j$ is uniquely determined by $k.$  The rest of row $m$ is then
defined by
\begin{equation*}
t(m,h)=\left\lfloor c^{j+h-1}/d^{k}\right\rfloor ,\text{ for }h=1,2,\ldots .
\end{equation*}

\bigskip

\textbf{Example 1.}  The array  $T_{(3,2,0)}$ consists of numbers $
\left\lfloor \frac{3^{j}}{2^{k}}\cdot 3^{h-1}\right\rfloor ,$ $
h=1,2,3,\ldots $
\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{9}{|c|}{\textbf{Table 1.  }$T_{(3,2,0)}$} \\ \hline
$1$ & $3$ & $9$ & $27$ & $81$ & $243$ & $729$ & $2187$ & $\cdots $ \\ \hline
$2$ & $6$ & $20$ & $60$ & $182$ & $546$ & $1640$ & $4900$ &  \\ \hline
$4$ & $13$ & $40$ & $121$ & $364$ & $1093$ & $3280$ & $9841$ &  \\ \hline
$5$ & $15$ & $45$ & $136$ & $410$ & $1230$ & $3690$ & $11071$ &  \\ \hline
$7$ & $22$ & $68$ & $205$ & $615$ & $1845$ & $5535$ & $16607$ &  \\ \hline
$8$ & $25$ & $76$ & $230$ & $691$ & $2075$ & $6227$ & $18683$ &  \\ \hline
$10$ & $30$ & $91$ & $273$ & $820$ & $2460$ & $7831$ & $22143$ &  \\ \hline
$\vdots $ &  &  &  &  &  &  &  &  \\ \hline
\end{tabular}
\end{equation*}
The rows of $T_{(3,2,0)}$, indexed by $m=1,2,3,\ldots ,$ are given by $
(j,k)=(0,0),$ then $(j,k)=(2,2),$ then $(j,k)=(2,1),\ldots ,$ as indicated
here:

\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{15}{|c|}{\textbf{Table 2.  The pairs }$(j,k)=(j_{m},k_{m})$
for $T_{(3,2,0)}$} \\ \hline
$m$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $
12 $ & $13$ & $14$ \\ \hline
$j$ & $0$ & $2$ & $2$ & $4$ & $5$ & $7$ & $4$ & $6$ & $8$ & $10$ & $12$ & $7$
& $14$ & $9$ \\ \hline
$k$ & $0$ & $2$ & $1$ & $4$ & $5$ & $8$ & $3$ & $6$ & $9$ & $12$ & $15$ & $7$
& $18$ & $10$ \\ \hline
\end{tabular}
\end{equation*}

Table 1 shows how an interspersion begets a fractal sequence:  for each $n,$
we write the number of the row containing $n$:
\begin{equation*}
(1,2,1,3,4,2,5,6,1,7,8,9,3,10,4,11,12,13,14,2,15,5,\ldots ),
\end{equation*}
a sequence which contains itself as a proper subsequence (infinitely many
times).

To conclude this introduction, we note that the arrays $T_{(c,d,k_{0})}$
represent a class of interspersions new to the literature.  A few
historical notes will help to place the topics of interspersions,
dispersions, and fractal sequences within a wider context.  Possibly the
earliest published array which is an interspersion was published by Kenneth
Stolarsky \cite{Stolarsky} with a revealing title, ``A set of generalized
Fibonacci sequences such that each natural number belongs to exactly
one''.
In 1980,
David Morrison introduced another interspersion, the Wythoff array.  Both
the Stolarksy and Wythoff arrays are presented in Neil Sloane's 
\textit{Classic Sequences} \cite{SloCS},
which also gives additional twentieth
century references, including \cite{Kim2}, where the terms 
``interspersion'' and 
``dispersion'' are introduced and proved equivalent, and
\cite{Kim3} in which fractal sequences are defined.  Twenty-first century
references include \cite{KimBrown,Kim}.

\section{Verification of interspersion properties}

\begin{lemma}
\label{lemma1}
Suppose $s/r$ is a positive irrational number and $
0<\delta <$ $\epsilon .$  Then there exist arbitrarily large integers $j$
and $k$ such that
\begin{equation}
\delta <jr-ks<\epsilon .
\label{eq1}
\end{equation}
\end{lemma}

\begin{proof}
First, suppose $\delta =0.$  Let $j_{i}/k_{i}$ be the $i$th
convergent to $s/r,$ so by \cite{Lang}, for all sufficiently large $i,$
we have
\begin{equation*}
|s/r-j_{i}/k_{i}|<1/k_{i}^{2}.
\end{equation*}
Let $i$ be large enough that $k_{i}>r/\epsilon $ and $j_{i}/k_{i}>s/r.$ 
Then
\begin{equation*}
|s/r-j_{i}/k_{i}|<\epsilon /rk_{i},
\end{equation*}
whence $0<j_{i}r-k_{i}s<\epsilon ,$ as desired.

Now suppose there exists $\delta >0$ such that for some $J$ and $K,$ the
inequality (\ref{eq1}) fails
for all $(j,k)$ satisfying $j\geq J$ and $k\geq K.$ 
Let $j^{\prime }$ and $k^{\prime }$ satisfy $ j^{\prime }\geq J,$ $
k^{\prime }\geq K,$ and
\begin{equation*}
0<j^{\prime }r-k^{\prime }s<\epsilon -\delta ,
\end{equation*}
and let $\delta _{1}=j^{\prime }r-k^{\prime }s.$  Then
\begin{equation*}
\epsilon /\delta _{1}-\delta /\delta _{1}>1,
\end{equation*}
so that
\begin{equation*}
\delta /\delta _{1}<q<\epsilon /\delta _{1}
\end{equation*}
for some $q\geq 1.$  Thus, taking $j=qj^{\prime }$ and $k=qk^{\prime },$ we
have $\delta <jr-ks<\epsilon ,$ a contradiction.  
\end{proof}

\begin{lemma}
Every $n$ can be represented as $\left\lfloor
c^{j}/d^{k}\right\rfloor $ using arbitrarily large $j$ and $k$.
\label{lemma2}
\end{lemma}

\begin{proof}
In Lemma~\ref{lemma1}, put $s=\ln c$ and $t=\ln d$; put $\delta =\ln
n $ and $\epsilon =\ln (n+1),$ and let $j$ and $k$ be arbitrarily large
integers satisfying (1):
\begin{equation*}
\ln n<j\ln c-k\ln d<\ln (n+1).
\end{equation*}
Equivalently, $n<c^{j}/d^{k}<n+1,$ so that $n=\left\lfloor
c^{j}/d^{k}\right\rfloor .$  
\end{proof}

\begin{lemma}
Suppose $n$ is a term in $T=T_{(c,d,x_{0})},$ so that $
n=t(m,h)$ for some $(m,h).$  Then the row-successor of $n$ is given by
\begin{equation*}
t(m,h+1)=cn+q\text{ for some }q\text{ satisfying }0\leq q\leq c-1.
\end{equation*}
\label{lemma3}
\end{lemma}

\begin{proof}
We have $n=\left\lfloor c^{j}/d^{k}\right\rfloor
=c^{j}/d^{k}-\delta ,$ where $0<\delta <1,$ so that $cn=c^{j+1}/d^{k}-c
\delta .$  Also, $t(m,h+1)=c^{j+1}/d^{k}-\epsilon ,$ where $0<\epsilon <1,$
so that
\begin{equation*}
t(m,h+1)-cn=c\delta -\epsilon .
\end{equation*}
Now $0<c\delta <c,$ so that $-1<c\delta -\epsilon <c.$  Because $c\delta
-\epsilon $ is an integer, we conclude that it is in $\{0,1,\ldots ,c-1\}.$
\end{proof}

\begin{lemma}
\label{lemma4}
No two terms of the array $T=T_{(c,d,k_{0})}$ are equal.
\end{lemma}

\begin{proof}
Suppose, to the contrary, that there are distinct terms $
n=\left\lfloor c^{j}/d^{k}\right\rfloor $ and $n_{1}=\left\lfloor
c^{j_{1}}/d^{k_{1}}\right\rfloor $ such that $n=n_{1}$.  Assume, without
loss of generality, that $j$ is the least exponent for which $\left\lfloor
c^{j_{1}}/d^{k_{1}}\right\rfloor =\left\lfloor c^{j}/d^{k}\right\rfloor $
for some $j_{1}$ and $k_{1}.\medskip $

\textit{Case 1}:  neither $n$ nor $n_{1}$ lies in column 1 of $T.$  By
Lemma \ref{lemma3},
\begin{equation*}
n=c\left\lfloor c^{j-1}/d^{k}\right\rfloor +q\text{ \ \ \ and \ \ }
n_{1}=c\left\lfloor c^{j_{1}-1}/d^{k_{1}}\right\rfloor +q_{1},
\end{equation*}
where $0\leq q\leq c-1$ and $0\leq q_{1}\leq c-1.$ \ Thus,
\begin{equation*}
c\left\lfloor c^{j-1}/d^{k}\right\rfloor +q=c\left\lfloor
c^{j_{1}-1}/d^{k_{1}}\right\rfloor +q_{1},
\end{equation*}
so that, assuming without loss that $\left\lfloor c^{j-1}/d^{k}\right\rfloor
\geq \left\lfloor c^{j_{1}-1}/d^{k_{1}}\right\rfloor ,$ we have
\begin{equation*}
\left\lfloor c^{j-1}/d^{k}\right\rfloor -\left\lfloor
c^{j_{1}-1}/d^{k_{1}}\right\rfloor =(q_{1}-q)/c.
\end{equation*}
But $0\leq (q_{1}-q)/c<1,$ so that, as $(q_{1}-q)/c$ is an integer, we have $
q_{1}=q$ and $\left\lfloor c^{j-1}/d^{k}\right\rfloor =\left\lfloor
c^{j_{1}-1}/d^{k_{1}}\right\rfloor ,$ contrary to the minimality of $
j$.

\bigskip

\textit{Case 2}:  one of the terms, $n$ or $n_{1},$ lies in column 1.  By
definition of column 1, $n$ and $n_{1}$ cannot both lie in column 1. 
Assume that $n$ but not $n_{1}$ lies in column 1.  Write $n=t(m,1)$ and $
n_{1}=t(m_{1},h),$ where $h\geq 2$.  Then by definition of $t(m,1),$ we
have $m_{1}\geq m,$ so that
\begin{equation*}
n\leq T\left( m_{1},1\right) <n_{1},
\end{equation*}
contrary to the assumption that $n=n_{1}.$ 
\end{proof}

\begin{theorem}
The array $T_{(c,d,k_{0})}$ is an interspersion.
\label{thm1}
\end{theorem}

\begin{proof}
By Lemma~\ref{lemma4}, property (I1) in the introduction holds, and
clearly (I2) and (I3) hold.  To see that (I4) holds, suppose
\begin{equation*}
t(m,h)<t(m^{\prime },h^{\prime })<t(m,h+1).
\end{equation*}
We must prove
\begin{equation*}
t(m,h+1)<t(m^{\prime },h^{\prime }+1)<t(m,h+2).
\end{equation*}
Since $t(m,h)<t(m^{\prime },h^{\prime }),$ we have $t(m^{\prime },h^{\prime
})-t(m,h)\geq 1,$ so that
\begin{equation*}
ct(m^{\prime },h^{\prime })-ct(m,h)\geq c.
\end{equation*}
Consequently, if $0\leq q_{1}\leq c-1$ and $0\leq q_{2}\leq c-1,$ then $
ct(m^{\prime },h^{\prime })-ct(m,h)\geq q_{1}-q_{2},$ so that
\begin{equation*}
ct(m,h)+q_{1}\leq ct(m^{\prime },h^{\prime })+q_{2},
\end{equation*}
which by Lemma~\ref{lemma3} implies $t(m,h+1)\leq t(m^{\prime },h^{\prime }+1),$ so
that by Lemma~\ref{lemma4},
\begin{equation*}
t(m,h+1)<t(m^{\prime },h^{\prime }+1)
\end{equation*}
Likewise, the inequality
\begin{equation*}
ct(m,h+1)-ct(m^{\prime },h^{\prime })\geq c
\end{equation*}
implies $t(m^{\prime },h^{\prime }+1)<t(m,h+2).$  
\end{proof}

\section{Permutations of $\Bbb N$}

Suppose $c,d,k_{0}$ are as already stipulated, and abbreviate $
T_{(c,d,k_{0})}$ as $T.$  In this section, we shall show that the exponents 
$k$ in the representation $\left\lfloor c^{j}/d^{k}\right\rfloor $ for the
numbers in $T$ form a permutation of the sequence ${\Bbb N} =(0,1,2,\ldots ).$
For example, as indicated in Table 2, for $(c,d,k_{0})=(3,2,0),$ the
sequence of values of $k$ is
\begin{equation*}
(0,2,1,4,5,8,3,6,9,12,15,7,18,10,\ldots ).
\end{equation*}

\begin{theorem}
\label{thm2}
Regarding the interspersion $T_{(c,d,k_{0})},$ let
\begin{equation*}
\left\lfloor (c^{j_{m}}/d^{k_{m}})c^{h-1}\right\rfloor ,\text{ for }
h=1,2,3,\ldots ,
\end{equation*}
be the numbers in row $m$. $ $Then each $n\geq k_{0}$ occurs exactly once
in the sequence $(k_{m})$.
\end{theorem}

\begin{proof}
Suppose, to the contrary, that there is a least $K\geq
k_{0}$ for which, for every $j,$
\begin{equation*}
\left\lfloor c^{j}/d^{K}\right\rfloor =\left\lfloor
c^{p_{j}}/d^{k}\right\rfloor 
\end{equation*}
for some $k$ satisfying $k_{0}\leq k<K$ and $p_{j}.$  Then
\begin{equation*}
\left\vert \frac{c^{j}}{d^{K}}-\frac{c^{p_{j}}}{d^{k}}\right\vert <1.
\end{equation*}
Moreover, as $k<K,$ we have $p_{j}<j$ and can write $K=k+e$ where $e>0$ and $
j=p_{j}+e_{j}$ where $e_{j}>0,$ so that
\begin{equation*}
\left\vert \frac{c^{e_{j}}}{d^{e}}-1\right\vert <\frac{d^{k}}{c^{^{p_{j}}}}.
\end{equation*}
As $j\rightarrow \infty ,$ clearly $p_{j}\rightarrow \infty ,$ so that $
\frac{d^{k}}{c^{^{p_{j}}}}\rightarrow 0.$  Consequently, $c^{e_{j}}=d^{e}$
for all sufficiently large $j,$ contrary to the independence of $c$ and $d,$
as defined and hypothesized in the introduction.  Thus, there is no such $
K,$ which is to say that for every $k\geq k_{0}$, there exists a row of $T$
such that the numbers in that row are the numbers $\left\lfloor
(c^{j}/d^{k})c^{h-1}\right\rfloor $ for some $j.$ By definition of $t(m,1)$
as the least $\left\lfloor c^{j}/d^{k}\right\rfloor =\left\lfloor
c^{j_{m}}/d^{k_{m}}\right\rfloor $ not in a row numbered $1,2,...,m-1$, the
numbers $k_{m}$ are distinct. 
\end{proof}

Regarding the set $\Bbb N$ of natural numbers to be $\{1,2,3,...\},$
Theorem~\ref{thm2}
shows that the sequence $(k_{m}-k_{0}+1)$ is a permutation of $\Bbb N$.  Do such
permutations have notable asymptotics?  Can they be efficiently computed?
We leave these questions open.

\section{Examples}

In Theorem~\ref{thm1}, the index $k_{0}$ can be any nonnegative integer, and in
Example 1, $k_{0}=0.$  In Table 3, we keep $(c,d)=(3,2)$ as in Table 1 but
change $k_{0}$ to $1$.  In infinitely many cases, a row of $T_{(3,2,0)}$ is
identical to a row of $T_{(3,2,1)},$ and in infinitely many cases a row of $
T_{(3,2,0)}$ is not identical to a row of $T_{(3,2,1)}$. These easily
proved observations remain true for $k_{0}=2,3,4,\ldots .$

\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{9}{|c|}{\textbf{Table 3.  }$T_{(3,2,1)}$} \\ \hline
$1$ & $4$ & $13$ & $40$ & $121$ & $364$ & $1093$ & $3280$ & $\cdots $ \\ 
\hline
$2$ & $6$ & $20$ & $60$ & $182$ & $546$ & $1640$ & $4920$ &  \\ \hline
$3$ & $10$ & $30$ & $91$ & $273$ & $820$ & $2460$ & $7381$ &  \\ \hline
$5$ & $15$ & $45$ & $136$ & $410$ & $1230$ & $3690$ & $11071$ &  \\ \hline
$7$ & $22$ & $68$ & $205$ & $615$ & $1845$ & $5535$ & $16607$ &  \\ \hline
$8$ & $25$ & $76$ & $230$ & $691$ & $2075$ & $6227$ & $18683$ &  \\ \hline
$9$ & $28$ & $86$ & $259$ & $778$ & $2335$ & $7006$ & $21018$ &  \\ \hline
$\vdots $ &  &  &  &  &  &  &  &  \\ \hline
\end{tabular}
\end{equation*}
The rows of $T_{(3,2,1)}$, indexed by $m=1,2,3,\ldots ,$ are given by $
(j,k)=(1,1),$ then $(j,k)=(2,2),$ then $(j,k)=(3,3),\ldots ,$ as indicated
here:

\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{15}{|c|}{\textbf{Table 4.  The pairs }$(j,k)=(j_{m},k_{m})$
for $T_{(3,2,1)}$} \\ \hline
$m$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $
12 $ & $13$ & $14$ \\ \hline
$j$ & $1$ & $2$ & $3$ & $4$ & $5$ & $7$ & $9$ & $6$ & $8$ & $10$ & $12$ & $7$
& $14$ & $9$ \\ \hline
$k$ & $1$ & $2$ & $3$ & $4$ & $5$ & $8$ & $11$ & $6$ & $9$ & $12$ & $15$ & $
7 $ & $18$ & $19$ \\ \hline
\end{tabular}
\end{equation*}
The fractal sequence corresponding to $T_{(3,2,1)}$ is
\begin{equation*}
(1,2,3,1,4,2,5,6,7,3,8,9,1,10,4,11,12,13,14,2,15,5,16,17,6,\ldots ).
\end{equation*}
Next, we change $k_{0}$ to $3:$
\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{9}{|c|}{\textbf{Table 5.  }$T_{(3,2,3)}$} \\ \hline
$1$ & $3$ & $10$ & $30$ & $91$ & $273$ & $820$ & $2460$ & $\cdots $ \\ \hline
$2$ & $7$ & $22$ & $68$ & $205$ & $615$ & $1845$ & $5535$ &  \\ \hline
$4$ & $12$ & $38$ & $115$ & $345$ & $1037$ & $3113$ & $9341$ &  \\ \hline
$5$ & $15$ & $45$ & $136$ & $410$ & $1230$ & $3690$ & $11071$ &  \\ \hline
$6$ & $19$ & $57$ & $172$ & $518$ & $1556$ & $4670$ & $14012$ &  \\ \hline
$8$ & $25$ & $76$ & $230$ & $691$ & $2075$ & $6227$ & $18683$ &  \\ \hline
$9$ & $28$ & $86$ & $259$ & $778$ & $2335$ & $7006$ & $21018$ &  \\ \hline
$\vdots $ &  &  &  &  &  &  &  &  \\ \hline
\end{tabular}
\end{equation*}
The rows of $T_{(3,2,3)}$, indexed by $m=1,2,3,\ldots ,$ are given by $
(j,k)=(2,3),$ then $(j,k)=(4,5),$ then $(j,k)=(7,9),\ldots ,$\ as indicated
here:

\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{15}{|c|}{\textbf{Table 6.  The pairs }$(j,k)=(j_{m},k_{m})$
for $T_{(3,2,3)}$} \\ \hline
$m$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $
12 $ & $13$ & $14$ \\ \hline
$j$ & $2$ & $4$ & $7$ & $4$ & $8$ & $7$ & $9$ & $6$ & $15$ & $10$ & $12$ & $
7 $ & $14$ & $16$ \\ \hline
$k$ & $3$ & $5$ & $9$ & $4$ & $10$ & $8$ & $11$ & $6$ & $20$ & $12$ & $15$ & 
$7$ & $18$ & $21$ \\ \hline
\end{tabular}
\end{equation*}

The fractal sequence corresponding to $T_{(3,2,3)}$ is
\begin{equation*}
(1,2,1,3,4,5,2,6,7,1,8,3,9,10,4,11,12,13,5,14,15,2,16,17,6,\ldots ).
\end{equation*}
As a final example, consider the interspersion $T_{(2,3,0)}:$
\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{9}{|c|}{\textbf{Table 7.  }$T_{(2,3,0)}$} \\ \hline
$1$ & $2$ & $4$ & $8$ & $16$ & $32$ & $64$ & $128$ & $\cdots $ \\ \hline
$3$ & $7$ & $14$ & $28$ & $56$ & $113$ & $227$ & $455$ &  \\ \hline
$5$ & $10$ & $21$ & $42$ & $85$ & $170$ & $341$ & $682$ &  \\ \hline
$6$ & $12$ & $25$ & $50$ & $101$ & $202$ & $404$ & $809$ &  \\ \hline
$9$ & $189$ & $37$ & $75$ & $151$ & $303$ & $606$ & $1213$ &  \\ \hline
$11$ & $22$ & $44$ & $89$ & $179$ & $359$ & $719$ & $1438$ &  \\ \hline
$13$ & $26$ & $53$ & $106$ & $213$ & $426$ & $852$ & $1704$ &  \\ \hline
$\vdots $ &  &  &  &  &  &  &  &  \\ \hline
\end{tabular}
\end{equation*}
The rows of $T_{(2,3,0)}$, indexed by $m=1,2,3,\ldots ,$ are given by $
(j,k)=(0,0),$ then $(j,k)=(5,2),$ then $(j,k)=(4,1),\ldots ,$\ as indicated
here:

\begin{equation*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{15}{|c|}{\textbf{Table 8.  The pairs }$(j,k)=(j_{m},k_{m})$
for $T_{(2,3,0)}$} \\ \hline
$m$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $
12 $ & $13$ & $14$ \\ \hline
$j$ & $0$ & $5$ & $4$ & $9$ & $8$ & $13$ & $18$ & $23$ & $20$ & $17$ & $44$
& $22$ & $30$ & $46$ \\ \hline
$k$ & $0$ & $2$ & $1$ & $4$ & $3$ & $6$ & $9$ & $12$ & $10$ & $8$ & $25$ & $
11$ & $16$ & $26$ \\ \hline
\end{tabular}
\end{equation*}

The fractal sequence corresponding to $T_{(2,3,0)}$ is
\begin{equation*}
(1,1,2,1,3,4,2,1,5,3,6,4,7,2,8,1,9,5,10,11,3,6,12,13,4,7,14,2,\ldots ).
\end{equation*}

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\textit{J. Integer Sequences} {\bf 7} (2004), Article 04.1.6.

\bibitem{Kim2} C. Kimberling, Interspersions and dispersions,
{\it Proc. Amer. Math. Soc.} {\bf 117} (1993), 313--321.

\bibitem{Kim3} C. Kimberling, Numeration systems and fractal sequences,
{\it Acta Arith.} {\bf 73} (1995), 103--117.

\bibitem{Kim} C. Kimberling, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling2/kimberling45.html}{The equation $(j+k+1)^{2}-4k=Qn^{2}$ and
related dispersions}, \textit{J. Integer Sequences} {\bf 10} (2007),
Article 07.2.7.

\bibitem{Lang} S. Lang, \textit{Introduction to Diophantine Approximations},
Addison-Wesley, Reading, Massachusetts, 1966.

\bibitem{SloOEIS} N. J. A. Sloane, editor, \href{http://www.research.att.com/~njas/sequences/index.html}{The On-Line Encyclopedia
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\href{http://www.research.att.com/~njas/sequences/index.html}{\tt http://www.research.att.com/\symbol{126}njas/sequences/}.

\bibitem{SloCS} N. J. A. Sloane, Classic Sequences In \textit{The On-Line
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\bibitem{Stolarsky} 
K. Stolarsky,
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Fibonacci sequences such that each natural number belongs to exactly one,
\textit{Fib. Quart.}, {\bf 15} (1977), 224.  

\bibitem{MathWorld1} E. Weisstein, \textit{MathWorld}, Fractal Sequence, \\
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\end{thebibliography}




\bigskip
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\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B99

\noindent \emph{Keywords: } interspersion, fractal sequence.

\bigskip
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\bigskip

\noindent 
(Concerned with sequences  
\seqnum{A007337},
\seqnum{A022447},
\seqnum{A114537},
\seqnum{A114577},
\seqnum{A120862},
\seqnum{A120863},
\seqnum{A124904},
\seqnum{A124905},
\seqnum{A124906},
\seqnum{A124907},
\seqnum{A124908},
\seqnum{A124909},
\seqnum{A124910},
\seqnum{A124911},
\seqnum{A124912},
\seqnum{A124913},
\seqnum{A124914},
\seqnum{A124915},
\seqnum{A124916},
\seqnum{A124917},
\seqnum{A124918},
\seqnum{A124919},
\seqnum{A125150},
\seqnum{A125151},
\seqnum{A125152},
\seqnum{A125153},
\seqnum{A125154},
\seqnum{A125155},
\seqnum{A125156},
\seqnum{A125157},
\seqnum{A125158},
\seqnum{A125159},
\seqnum{A125160}, and
\seqnum{A125161}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received December 30 2006;
revised version received  May 4 2007.
Published in {\it Journal of Integer Sequences}, May 6 2007.

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\noindent
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\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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