Categorification of the Fibonacci Numbers Using Representations of Quivers

Philipp Fahr Brehmstr. 51 D-40239 Düsseldorf Germany

                   philfahr@gmx.net

Claus Michael Ringel Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 P. R. China and King Abdulaziz University P. O. Box 80200 Jeddah Saudi Arabia

             ringel@math.uni-bielefeld.de

Abstract

In a previous paper we presented a partition formula for the even-index Fibonacci numbers using the preprojective representations of the 3-Kronecker quiver and its universal cover, the 3-regular tree. In this paper we deal in a similar way with the oddindex Fibonacci numbers. The Fibonacci modules introduced here provide a convenient categorification of the Fibonacci numbers.

1 Introduction

We consider the Fibonacci numbers with f_{0} = 0, f_{1} = 1 and the recursion rule f_{i+1} = f_{i}+f_{i−1} for i ≥ 1. As suggested by many authors, one may use this recursion rule also for i ≤ 0 and one obtains in this way Fibonacci numbers f_{i} for all integral indices i (Knuth [11] calls this extended set the negaFibonacci numbers); the sequence f_{t} with −10 ≤ t ≤ 10 looks as follows:

−55, 34, −21, 13, −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55;

in general, f_{−t} = (−1)^{t+1}f_{t}. This rule, but also many other observations, show that Fibonacci numbers have quite different properties depending whether the index is even or odd, and the present report will strongly support this evidence.

Our aim is to outline a categorification of the Fibonacci numbers, or better of Fibonacci pairs, using the representation theory of quivers. By definition, Fibonacci pairs are the pairs of the form [f_{t}, f_{t+2}] or [f_{t}, f_{t−2}]; such a pair will be called even or odd provided the index t of f_{t} is even or odd, respectively.

It is well-known that the dimension vectors of some indecomposable representations of the 3-Kronecker quiver Q are Fibonacci pairs. In our previous note [6], preprojective and preinjective representations were used in order to derive a partition formula for the evenindex Fibonacci numbers f_{2n}. In the present paper we will exhibit a corresponding partition formula for the odd-index Fibonacci numbers f_{2n+1}. Now regular representations of Q play a role (see also Chen [3] and Ringel [14]). Actually, we will be dealing directly with the universal cover of Q, thus with graded representations of Q. The universal cover of Q is the quiver (T, E, Ω), where (T, E) is the 3-regular tree and Ω is a bipartite orientation. In section 4 we will introduce some representations P_{t}(x) and R_{t}(x, y) of the quiver (T, E, Ω) which we label Fibonacci modules. The Fibonacci modules will be used in section 4 and 6 in order to categorify the Fibonacci pairs (and hence the Fibonacci numbers). The dimension vectors of the Fibonacci modules are called Fibonacci vectors, they are studied already in section 2 and 3 and provide the partition formula.

Categorification of a set of numbers means to consider instead of these numbers suitable objects in a category (here representations of quivers), so that the numbers in question occur as invariants of the objects (see for example Crane and Yetter [4]). Equality of numbers may be visualized by isomorphisms of objects, functional relations by functorial ties. We will see that certain addition formulas for Fibonacci numbers can be interpreted very well by displaying filtrations of Fibonocci modules.

The partition formula for the odd-index Fibonacci numbers presented here is due to Fahr [5], the remaining considerations are based on discussions of the authors during the time when Fahr was a PhD student at Bielefeld, the final version was written by Ringel. In a forthcoming paper [7] we will show that the numbers which are used in the partition formulae can be arranged quite well in two triangles which we call the Fibonacci partition triangles. They turn out to be quite similar to the Pascal triangle for the binomial coefficients and are strongly interrelated.

2 Fibonacci Vectors

Let us start with the 3-regular tree (T, E) with vertex set T and edge set E. We consider paths in this tree, thus sequences x_{0}, x_{1}, …, x_{t} of elements x_{i} of T such that all the sets {x_{i−1}, x_{i}} belong to E, where 1 ≤ i ≤ t, and x_{i+1} ≠ x_{i−1} for 0 < i < t; if such a sequence exists, then we say that x_{0} and x_{t} have distance t and write d(x_{0}, x_{t}) = t. As usual, vertices with distance 1 are said to be neighbors.

Let K_{0}(T) be the free abelian group generated by the vertex set T, or better by elements s(x) corresponding bijectively to the vertices x ∈ T. The elements of K_{0}(T) can be written as finite sums a = ∑_{x∈T} a_{x}s(x) with a_{x} ∈ ℤ (and almost all a_{x} = 0), or just as a = (a_{x})_{x∈T}; the set of vertices x with a_{x} ≠ 0 will be called the support of a.

We are going to single out suitable elements of K_{0}(T) which we will call Fibonacci vectors: For every pair (x, y) with {x, y} ∈ E, let r(x, y) = s(x) + s(y). The Fibonacci vectors will be obtained from the elements s(x) and r(x, y) by applying sequences of reflections. For y ∈ T the reflection σ^{y} : K_{0}(T) → K_{0}(T) is defined as follows: For a ∈ K_{0}(T), its image σ^{y}a is given by (σ^{y}a)_{x} = a_{x} for x ≠ y, and (σ^{y}a)_{y} = −a_{y} +∑_{{x,y}∈E} a_{x}. Observe that the reflections σ^{y}, σ^{y′} commute provided y, y′ are not neighbors.

We denote by Σ^{x} the composition of all the reflections σ^{y} with d(x, y) being even (note that this composition is defined: first of all, since any element a ∈ K_{0}(T) has finite support, the number of vertices y with σ^{y}a ≠ a is finite; second, given two different vertices y, y′ such that the distance between x and y, as well as between x and y′ is even, then y, y′ cannot be neighbors, thus σ^{y}, σ^{y′} commute). Similarly, we denote by \underline{Σ}^{x} the composition of all the reflections σ^{y} with d(x, y) being odd (again, this is well-defined).

We define s_{t}(x) for all t ≥ 0, using induction: First of all, s_{0}(x) = s(x). If s_{t}(x) is already defined for some t ≥ 0, let s_{t+1}(x) = \underline{Σ}^{x}s_{t}(x) in case t is even, and s_{t+1}(x) = Σ^{x}s_{t}(x) in case t is odd, thus

s_{0}(x) = s(x),
s_{1}(x) = \underline{Σ}^{x}s(x),
s_{2}(x) = Σ^{x}\underline{Σ}^{x}s(x),
s_{3}(x) = \underline{Σ}^{x}Σ^{x}\underline{Σ}^{x}s(x),
…

Similarly, we define r_{t}(x, y) for all t ≥ 0: First of all, r_{0}(x, y) = r(x, y). If r_{t}(x, y) is already defined for some t ≥ 0, let r_{t+1}(x, y) = \underline{Σ}^{x}r_{t}(x, y) in case t is even, and r_{t+1}(x, y) = Σ^{x}r_{t}(x, y) in case t is odd. The elements s_{t}(x) with t ≥ 0 as well as the elements r_{t}(x, y) with t ∈ ℤ will be called the Fibonacci vectors.

Define

s_{t}(x)_{−} = ∑_{d(x,z)≢t} s_{t}(x)_{z}, and s_{t}(x)_{+} = ∑_{d(x,z)≡t} s_{t}(x)_{z},

where ≡ means equivalence modulo 2, and similarly,

r_{t}(x, y)_{−} = ∑_{d(x,z)≢t} r_{t}(x, y)_{z}, and r_{t}(x, y)_{+} = ∑_{d(x,z)≡t} r_{t}(x, y)_{z}.

Proposition 1. The Fibonacci vectors have non-negative coordinates and

s_{t}(x)_{−} = f_{2t}, s_{t}(x)_{+} = f_{2t+2}, r_{t}(x, y)_{−} = f_{2t−1}, r_{t}(x, y)_{+} = f_{2t+1}.

Proof. The proof for s_{t} has been given in a previous paper [6]. In the same way, one deals with r_{t}(x, y). But note that in the previous paper [6] we have fixed some base point x_{0}, and we wrote Φ_{0}, Φ_{1} instead of Σ^{x_{0}}, \underline{Σ}^{x_{0}} respectively. Also, there, we have denoted the function s_{2t}(x_{0}) by a_{t}, and the functions s_{2t+1}(x_{0}) by Φ_{1}a_{t}.

The functions s_{t}(x_{0}) have been exhibited in the previous paper [6], for 0 ≤ t ≤ 5. At the end of the present paper we present the functions r_{t}(x, y) for 0 ≤ t ≤ 5.

3 The Partition Formula

In the previous paper [6] the functions s_{t}(x) were used in order to provide a partition formula for the even-index Fibonacci numbers (see also Prodinger [13] and Hirschhorn [9]). We now consider the functions r_{t}(x, y) in order to obtain a corresponding partition formula for the odd-index Fibonacci numbers. In order to get all odd-index Fibonacci numbers, it is sufficient to look at the functions r_{2t}(x, y), since

f_{4t−1} = r_{2t}(x, y)_{−}, f_{4t+1} = r_{2t}(x, y)_{+}.

For any t, the numbers s_{t}(x)_{z} only depend on the distance d(x, z). In contrast, for r_{t}(x, y) and d(x, z) > 0, there are now two values which occur as r_{t}(x, y) depending on whether the path connecting x and z runs through y or not. Let us denote by T_{s}(x, y) the set of vertices z of T with d(x, z) = s and such that the path connecting x, z does not involve y, and T′_{s}(x, y) shall denote the set of vertices z of T with d(x, z) = s, such that the path connecting x, z does involve y. Clearly, we have for s ≥ 1

|T_{s}(x, y)| = 2^{s},  |T′_{s}(x, y)| = 2^{s−1}.  (1)

The proof of (1) is by induction: Of course, T′_{1}(x, y) = {y} whereas T_{1}(x, y) consists of the two remaining neighbors of x. For any vertex z with d(x, z) = s > 0, there are precisely two neighbors z′, z′′ such that d(x, z′) = d(x, z′′) = s + 1, therefore |T_{s+1}(x, y)| = 2|T_{s}(x, y)| and |T′_{s+1}(x, y)| = 2|T′_{s}(x, y)|.

Thus, let us look at the function u_{t} : ℤ → ℕ defined for s ≥ 0 by u_{t}[s] = r_{2t}(x, y)_{z} with z ∈ T_{s}(x, y), and for s ≤ −1 by u_{t}[s] = r_{2t}(x, y)_{z} with z ∈ T′_{−s}(x, y). Using the equality (1), we see:

f_{4t−1} = r_{2t}(x, y)_{−} = ∑_{d(x,z) odd} r_{2t}(x, y)_{z}
= ∑_{s≥0 odd} |T_{s}(x, y)|u_{t}[s] + ∑_{s>0 odd} |T′_{s}(x, y)|u_{t}[−s]
= ∑_{s≥0 odd} 2^{s}u_{t}[s] + ∑_{s>0 odd} 2^{s−1}u_{t}[−s]

f_{4t+1} = r_{2t}(x, y)_{+} = ∑_{d(x,z) even} r_{2t}(x, y)_{z}
= ∑_{s≥0 even} |T_{s}(x, y)|u_{t}[s] + ∑_{s>0 even} |T′_{s}(x, y)|u_{t}[−s]
= ∑_{s≥0 even} 2^{s}u_{t}[s] + ∑_{s>0 even} 2^{s−1}u_{t}[−s]

Without reference to r_{2t}, we can define the functions u_{t} directly as follows: We start with u_{0}[0] = u_{0}[−1] = 1 and u_{0}[s] = 0 for all other s. If u_{t} is already defined for some t ≥ 0, then we define first u_{t+1}[s] for odd integers s by the rule

u_{t+1}[s] = \begin{cases}2u_{t}[s − 1] − u_{t}[s] + u_{t}[s + 1], & if s < 0; \\ u_{t}[s − 1] − u_{t}[s] + 2u_{t}[s + 1], & if s > 0; \end{cases}

and in a second step for even integers s by

u_{t+1}[s] = \begin{cases}2u_{t+1}[s − 1] − u_{t}[s] + u_{t+1}[s + 1], & if s < 0; \\ u_{t+1}[s − 1] − u_{t}[s] + 2u_{t+1}[s + 1], & if s ≥ 0; \end{cases}

Here is the table of the numbers u_{t}[s], for t ≤ 4; in addition, we list the corresponding Fibonacci numbers f_{−1}, f_{1}, f_{3}, …, f_{17}.

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Remark 2. We can reformulate the recursion rules using the generalized Cartan matrix A = (a_{ij})_{ij} (indexed over the integers) with Dynkin diagram

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One pair of vertices is simply laced, for the remaining laced pairs, one of the numbers a_{ij}, a_{ji} is equal to −1, the other to −2 (as in the case of the Cartan matrix of type 𝔹_{2}). In order to obtain u_{t+1} from u_{t}, apply first the reflections σ_{i} with i odd, then the reflections σ_{i} with i even.

4 Fibonacci Modules

We now endow (T, E) with a bipartite orientation Ω; in this way, we deal with a quiver (T, E, Ω) and we consider its representations. Note that there are just two bipartite orientations for (T, E). Given a vertex x of T, let us denote by Ω_{t}^{x} the bipartite orientation such that x is a sink in case t is even, and a source, in case t is odd. Let P_{t}(x) be the indecomposable representation of (T, E, Ω_{t}^{x}) with dimension vector s_{t}(x). If there is an edge between x and y, let R_{t}(x, y) be the indecomposable representation of (T, E, Ω_{t}^{x}) with dimension vector r_{t}(x, y); it is well-known that these modules exist and are unique up to isomorphism: for t = 0 this assertion is trivial, and the Bernstein-Gelfand-Ponomarev reflection functors [2] can be used in order to construct P_{t+1}(x) from P_{t}(x), as well as R_{t+1}(x, y) from R_{t}(x, y).

We call the modules P_{t}(x) and R_{t}(x, y) Fibonacci modules.

Proposition 3. Let x be a vertex of T with neighbors y, y′, y′′ and t ≥ 1 an integer. Assume that for even t, the vertex x is a sink, and if t is odd, x is a source. Then there are exact sequences

0 → P_{t−1}(y) → P_{t}(x) → R_{t}(x, y) → 0,
0 → P_{t−1}(y′) → R_{t}(x, y) → R_{t−1}(y′′, x) → 0,

and they are unique up to isomorphism.

Proof. This is clear for t = 1 and follows by induction using the Bernstein-Gelfand-Ponomarev reflection functors.

Corollary 4. Let x_{0}, x_{1}, …, x_{t} be a path with x_{0} a sink. Then P_{t}(x_{t}) has a filtration

P_{0}(x_{0}) ⊂ P_{1}(x_{1}) ⊂ ⋯ ⊂ P_{t}(x_{t})

with factors

P_{i}(x_{i})/P_{i−1}(x_{i−1}) = R_{i}(x_{i}, x_{i−1})

for 1 ≤ i ≤ t.

Proof. Since x_{0} is a sink, we see that x_{i} is a sink for i even and a source for i odd. Thus, the proposition provides exact sequences

0 → P_{i−1}(x_{i−1}) → P_{i}(x_{i}) → R_{i}(x_{i}, x_{i−1}) → 0,

for 1 ≤ i ≤ t.

Corollary 5. Let x_{−1}, x_{0}, x_{1}, …, x_{t}, x_{t+1} be a path with x_{0} a source. For 0 ≤ i ≤ t, let z_{i} be the neighbor of x_{i} different from x_{i−1} and x_{i+1}. Then there is an exact sequence

0 → P_{0}(z_{0}) ⊕ ⋯ ⊕ P_{t}(z_{t}) → R_{t+1}(x_{t}, x_{t+1}) → R_{0}(x_{−1}, x_{0}) → 0.

Proof. Here, x_{0} is a source, thus we see that x_{i} is a source for i even and a sink for i odd. The proof is by induction on t, the case t = 0 should be clear. Assume that t ≥ 1. We consider the vertices x_{0}, …, x_{t+1}, z_{1}, … z_{t}, but deal with the opposite orientation Ω′ (so that now x_{1} is a source); the corresponding representations will be distinguished by a dash, thus P′_{i}(z_{i}) is the indecomposable representation of (T, E, Ω′) with dimension vector s_{i}(z_{i}), and so on. By induction, there is an exact sequence

0 → P′_{0}(z_{1}) ⊕ ⋯ ⊕ P′_{t−1}(z_{t}) → R′_{t}(x_{t}, x_{t+1}) → R′_{0}(x_{0}, x_{1}) → 0.

Applying reflection functors (at all sinks), we obtain the exact sequence

0 → P_{1}(z_{1}) ⊕ ⋯ ⊕ P_{t}(z_{t}) → R_{t+1}(x_{t}, x_{t+1}) → R_{1}(x_{0}, x_{1}) → 0.

Note that P_{0}(z_{0}) is a subrepresentation of R_{1}(x_{1}, x_{0}), thus we obtain the induced exact sequence

0 → P_{1}(z_{1}) ⊕ ⋯ ⊕ P_{t}(z_{t}) → W → P_{0}(z_{0}) → 0,

where W is a subrepresentation of R_{t+1}(x_{t}, x_{t+1}) with R_{t+1}(x_{t}, x_{t+1})/W being isomorphic to R_{1}(x_{1}, x_{0})/P_{0}(z_{0}) = R_{0}(x_{−1}, x_{0}).

On the other hand, the induced sequence has to split, since P_{0}(z_{0}) is projective, thus W is the direct sum of the representations P_{i}(z_{i}) with 0 ≤ i ≤ t. This completes the proof.

Of course, proposition 3 should be seen as a categorification of the defining equation

f_{t+1} = f_{t−1} + f_{t}

for the Fibonacci numbers. Corollaries 4 and 5 are categorifications of the equalities

f_{2t} = ∑_{i=1}^{t} f_{2i−1}, f_{2t+1} = 1 + ∑_{i=1}^{t} f_{2i}.

Remark 6. The representations R_{0}(x_{−1}, x_{0}), P_{0}(z_{0}), …, P_{t}(z_{t}) are pairwise orthogonal bricks, the corresponding Ext-quiver is

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Let F denote the full subcategory of of all representations of (T, E, Ω) with a filtration with factors of the form R_{0}(x_{−1}, x_{0}), P_{0}(z_{0}), …, P_{t}(z_{t}), then corollary 5 shows that R_{t+1}(x_{t}, x_{t+1}) is indecomposable projective in F.

5 Fibonacci Pairs

The Fibonacci pairs are related to the integral quadratic form q(x, y) = x^{2} + y^{2} − 3xy as follows:

Proposition 7. A pair [x, y] ∈ ℤ^{2} is a Fibonacci pair if and only if |q(x, y)| = 1.

The pairs [x, y] with |q(x, y)| = 1 form two hyperbolas. The hyperbola of all pairs [x, y] with q(x, y) = 1 yields the even Fibonacci pairs, these are the black dots on the following picture. The hyperbola q(x, y) = −1 yields the odd Fibonacci pairs, they are indicated by small circles.

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 x y

The quadratic form q on ℤ^{2} defines the Kac-Moody root system [10] for the Cartan

matrix [ \begin{matrix}2 & −3 \\ −3 & 2 \end{matrix} ] (see [6], section 6). The pairs [x, y] with q(x, y) = 1 are the real roots,

those with q(x, y) ≤ 0 the imaginary roots. We identify ℤ^{2} with the Grothendieck group of the 3-Kronecker modules, and q is the Euler form. Given a 3-Kronecker module M, the corresponding element in the Grothendieck group is its dimension vector dim M.

6 Recursion formula for Fibonacci numbers with index of the same parity

This is the recursion formula:

f_{t+2} = 3f_{t} − f_{t−2}.  (1)

Proof. The term f_{t+2} − 3f_{t} + f_{t−2} is just the sum of f_{t+2} − f_{t+1} − f_{t}, of f_{t+1} − f_{t} − f_{t−1} and of −(f_{t} − f_{t−1} − f_{t−2}), thus equal to zero.

Note that the recursion formula can also be rewritten as

f_{t−2} = 3f_{t} − f_{t+2},  (2)

thus the same recurrence formula works both for going up and for going down. Also, we can write it in the form

f_{t−2} + f_{t+2} = 3f_{t}.  (3)

The formulae (1) and (2) are categorified by looking at the reflection functors σ_{+} and σ_{−}: Given a 3-Kronecker module M = (M_{1}, M_{2}, α, β, γ), we form σ_{+}M by taking (σ_{+}M)_{2} = M_{1} and (σ_{+}M)_{1} as the kernel of the map (α, β, γ) : M_{1}^{3} → M_{2}; note that if M is indecomposable and not the simple projective module P_{0}, then this kernel has dimension 3 dim M_{1} − dim M_{2}. Similarly, we take (σ_{−}M)_{1} = M_{2} and (σ_{+}M)_{2} is by definition the cokernel of the map (α, β, γ)^{t} : M_{1} → M_{2}^{3}; this cokernel has dimension 3 dim M_{2} − dim M_{1} provided M is indecomposable and not the simple injective module Q_{0}.

Let us now look at the formula (3). Let P_{n} = (σ_{−})^{n}P_{0} and Q_{n} = (σ_{+})^{n}Q_{0}, with n ∈ ℕ. Also note that the indecomposable 3-Kronecker modules with dimension vector [1, 1] are indexed by the elements λ of the projective plane ℙ_{2}(k), such a module will be denoted by R_{0,λ} and we form R_{n,λ} = (σ_{+})^{n}R_{0,λ} and R_{−n,λ} = (σ_{−})^{n}R_{0,λ}, again with n ∈ ℕ. Of course, the dimension vectors of these modules are just Fibonacci pairs:

dim P_{n} = [f_{2n}, f_{2n+2}], dim Q_{n} = [f_{2n+2}, f_{2n}], dim R_{m,λ} = [f_{2m+1}, f_{2m−1}],

for all n ∈ ℕ, m ∈ ℤ. Let us consider the corresponding Auslander-Reiten sequences (see for example [1]), they are of the form

0 → P_{n−1} → P_{n}^{3} → P_{n+1} → 0,
0 → Q_{n+1} → Q_{n}^{3} → Q_{n−1} → 0,
0 → R_{m+1,λ} → E_{m,λ} → R_{m−1,λ} → 0,

with suitable indecomposable modules E_{m,λ} having dimension vector equal to 3 dim R_{m,λ}. We see that these Auslander-Reiten sequences categorify the formula (3).

7 Illustrations

The following illustrations show the functions r_{t}(x, y) for t = 0, .., 5, with x the center and y above x. The dotted circles indicate a fixed distance to x. We show (T, E) with bipartite orientation such that x is a sink (thus z is a sink if d(x, y) is even, and a source if d(x, z) is odd). The region containing the arrow y → x has been dotted. For the convenience of the reader we have indicated the corresponding Fibonacci pair [f_{2t−1}, f_{2t+1}].

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_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 1 11111  1 1 1 1 1 1 00 0 0 0 0 0 0  0 0 0 0 0 0 0 0} r_{0}(x, y) [1, 1] r_{1}(x, y) [1, 2] r_{2}(x, y) [2, 5] r_{3}(x, y) [5, 13] r_{4}(x, y) [13, 34] r_{5}(x, y) [34, 89]

10

References

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[2] I. N. Bernstein, I. M. Gelfand, and V. A. Ponomarev, Coxeter functors and Gabriel’s theorem, Russian Math. Surveys 29 (1973), 17–32.

[3] B. Chen, Regular modules and quasi-lengths over 3-Kronecker quiver using Fibonacci numbers, Arch. Math. 95 (2010), 105–113.

[4] L. Crane and D. N. Yetter, Examples of categorification, Cahiers Topologie Géom. Différentielle Catég. 39 (1998), 3–25.

[5] Ph. Fahr, Infinite Gabriel-Roiter Measures for the 3-Kronecker Quiver, Dissertation, Universität Bielefeld, 2008.

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[7] Ph. Fahr and C. M. Ringel, The Fibonacci partition triangles, preprint,

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[8] P. Gabriel, The universal cover of a representation-finite algebra, in M. Auslander and E. Lluis, eds., Representations of Algebras, Springer Lecture Notes in Math., Vol. 903, 1981, pp. 68–105.

[9] M. D. Hirschhorn, On recurrences of Fahr and Ringel arising in graph theory, J. Integer Sequences 12 (2009), Paper 09.6.8.

[10] V. G. Kac, Infinite-Dimensional Lie Algebras. 3rd edition, Cambridge University Press, 1990.

[11] D. Knuth, Negafibonacci numbers and the hyperbolic plane. San Jose Meeting of the Mathematical Association of America (2008),

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[12] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, New York, 2001.

[13] H. Prodinger, Generating functions related to partition formulae for Fibonacci numbers, J. Integer Sequences, 11 (2008), Paper 08.1.8.

[14] C. M. Ringel, Indecomposable representations of the Kronecker quivers, Proc. Amer. Math. Soc., to appear.

[15] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org.

2010 Mathematics Subject Classification: Primary 11B39; Secondary 16G20, 05A17, 05C05, 16G70. Keywords: Fibonacci number, quiver representation, universal cover, 3-regular tree, 3-Kronecker quiver, categorification.

(Concerned with sequences \underline{A000045}, \underline{A132262}, \underline{A147316}, \underline{A197956}, \underline{A197957}.)

Received May 26 2011; revised version received October 17 2011; December 29 2011. Published in Journal of Integer Sequences, December 29 2011.

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