Some results on average of Fibonacci and Lucas sequences

Optimal cube factors of Fibonacci and matchable Lucas cubes

The optimal cube factor of a graph, a special kind of component factor, is first introduced. Furthermore, the optimal cube factors of Fibonacci and matchable Lucas cubes are studied; and some results on the Padovan sequence and binomial coefficients are obtained.

Asymptotic Properties of Fibonacci Cubes and Lucas Cube

It is proved that the asymptotic average eccentricity and the asymptotic average degree of Fibonacci cubes and Lucas cubes are $(5+\sqrt 5)/10$ and $(5-\sqrt 5)/5$, respectively. A new labeling of the leaves of Fibonacci trees is introduced and proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. Hypercube density is also introduced and studied. The hypercube density of both Fibonacci cubes and Lucas cubes is shown to be $(1-1/\sqrt 5)/\log_2φ$, where $φ$ is the golden ratio, and the Cartesian produ…

The Mostar index of Fibonacci and Lucas cubes

The Mostar index of a graph was defined by Došlić, Martinjak, Škrekovski, Tipurić Spužević and Zubac in the context of the study of the properties of chemical graphs. It measures how far a given graph is from being distance-balanced. In this paper, we determine the Mostar index of two well-known families of graphs: Fibonacci cubes and Lucas cubes.

涉及斐波那契数与卢卡斯数的二项式和关系

Binomial sum relations involving Fibonacci and Lucas numbers

该研究探讨了含（广义）斐波那契数、卢卡斯数与不同类型二项式系数的二项式和的关联，给出多类二项式和关系并重新发现部分已作为问题提案的关系式。

Novel Results of Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals

This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type $_2F_{1}(z)$, for certain $z$. Several new connection formulae between some famous polynomials such as Fibonacci, Lucas, Pell, Fermat, Pell-Lucas, and Fermat-Lucas polynomials are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some…

Maximal hypercubes in Fibonacci and Lucas cubes

The Fibonacci cube $Γ_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $Λ_n$ is obtained from $Γ_n$ by removing vertices that start and end with 1. We characterize maximal induced hypercubes in $Γ_n$ and $Λ_n$ and deduce for any $p\leq n$ the number of maximal $p$-dimensional hypercubes in these graphs.

Analogues of a Fibonacci-Lucas Identity

Sury's 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn's 2003 book: {\em Proofs that count: The art of combinatorial proof}) has excited a lot of comment. We give an alternate, telescoping, proof of this---and associated---identities and generalize them. We also give analogous identities for other sequences that satisfy a three-term recurrence relation.

New binomial Fibonacci sums

We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.

Some properties of Fibonacci numbers, Fibonacci octonions and generalized Fibonacci-Lucas octonions

In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.

关于一个 Fibonacci–Lucas 恒等式的若干注记

Some notes on a Fibonacci-Lucas identity

本文围绕 Edgar–Bhatnagar 的 Fibonacci–Lucas 多项式恒等式，为广义 Lucas 序列提出新的多项式恒等式、特例及 Chebyshev 多项式推广。

Sums of Powers of Fibonacci and Lucas Polynomials in terms of Fibopolynomials

We study sums of powers of Fibonacci and Lucas polynomials of the form $% \sum_{n=0}^{q}F_{tsn}^{k}(x) $ and $\sum_{n=0}^{q}L_{tsn}^{k}% (x) $, where $s,t,k$ are given natural numbers, together with the corresponding alternating sums $\sum_{n=0}^{q}(-1) ^{n}F_{tsn}^{k}(x) $ and $\sum_{n=0}^{q}(-1) ^{n}L_{tsn}^{k}(x) $. We give sufficient conditions on the parameters $s,t,k$ for express these sums as linear combinations of certain $s$-Fibopolynomials.

Routing linear permutations on Fibonacci and Lucas cubes

In recent years there has been much interest in certain subcubes of hypercubes, namely Fibonacci cubes and Lucas cubes (and their generalized versions). In this article we consider online routing of linear permutations on these cubes. The model of routing we use regards edges as bi-directional, and we do not allow queues of length greater than one. Messages start out at different vertices, and in movements synchronized with a clock, move to an adjacent vertex or remain where they are, so that at the next stage there is still exactly one message per vertex. This is the routing model we defined …

Some beautiful q-analogues of Fibonacci and Lucas polynomials

We give an overview about well-known basic properties of two classes of q-Fibonacci and q-Lucas polynomials and offer a common generalization.

Some elementary aspects of q-Fibonacci and q-Lucas polynomials

Based on well-known properties of Fibonacci and Lucas numbers and polynomials we give a self-contained approach to some bivariate analogs.

Fibonacci 立方体与 Lucas 立方体中的边一般位置集

Edge general position sets in Fibonacci and Lucas cubes

本文研究偏立方体的边一般位置集，证明 Fibonacci 立方体与 Lucas 立方体中两个最大 Θ-类的并集构成极大边一般位置集。

Some remarks on generalized Fibonacci and Lucas polynomials

Starting with some determinants of binomial coefficients which are related to Fibonacci and Lucas polynomials we study similar determinants for some generalizations of these polynomials and their q-analogues.

广义 Fibonacci 数、Catalan 数与调和数的恒等式及生成函数

Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers

本文研究了广义 Fibonacci 与 Lucas 数类的性质，获得了著名 Fibonacci 恒等式的类比及乘积卷积新恒等式，并写下了其乘积、多截、调和数与 Catalan 数的生成函数。

斐波那契多项式与卢卡斯多项式的若干幂级数推广

Some power series extensions of Fibonacci and Lucas polynomials

本研究聚焦形式幂级数领域，探讨可作为偶/奇下标斐波那契、卢卡斯多项式插值的形式幂级数，拓展了相关多项式的幂级数研究方向。

Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.

Complementary Families of the Fibonacci-Lucas Relations

In this paper we present two families of Fibonacci-Lucas identities, with the Sury's identity being the best known representative of one of the families. While these results can be proved by means of the basic identity relating Fibonacci and Lucas sequences we also provide a bijective proof. Both families are then treated by generating functions.

关于由k-斐波那契数和k-卢卡斯数乘积定义的矩阵的一些性质

On Some Properties of Matrices with Entries Defined by Products of $k$-Fibonacci and $k$-Lucas Numbers

本文研究了由k-斐波那契数和k-卢卡斯数乘积定义的矩阵家族，推导了多个矩阵不变量的显式公式，并探讨了与整数序列的关系。

Sums of products of generalized Fibonacci and Lucas numbers

In this paper, we establish several formulae for sums and alternating sums of products of generalized Fibonacci and Lucas numbers. In particular, we recover and extend all results of Z. Cerin and Z. Cerin & G. M. Gianella, more easily.

Right Circulant Matrices with Generalized Fibonacci and\ Lucas Polynomials and Coding Theory

In this paper, we give two new coding algorithms by means of right circulant matrices with elements generalized Fibonacci and Lucas polynomials. For this purpose, we study basic properties of right circulant matrices using generalized Fibonacci polynomials $F_{p,q,n}\left( x\right) $, generalized Lucas polynomials $L_{p,q,n}\left( x\right) $ and geometric sequences.

非周期单铺块超铺块中的斐波那契与卢卡斯序列

Fibonacci and Lucas Sequences in Aperiodic Monotile Supertiles

本文研究帽子超铺块的大小与方向，发现其涉及斐波那契序列、卢卡斯序列及一个线性相关的整数序列，并将结果推广到帽子家族的所有非周期铺块。

Vertex and edge orbits of Fibonacci and Lucas cubes

The Fibonacci cube $Γ_n$ is obtained from the $n$-cube $Q_n$ by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube $Λ_n$ is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of the sizes of the vertex orbits of $Λ_n$ is $\{k \ge 1;\ k \divides n\} \cup\, \{k \ge 18;\ k \divides 2n\}$, the number of the vert…

Fibonacci-run 图与 Lucas-run 图的距离立方体多项式

Distance cube polynomials of Fibonacci and Lucas-run graphs

本文纠正了 Fibonacci-run 图距离立方体多项式生成函数的错误，并证明了建立 Lucas-run 图与 Fibonacci-run 图立方体多项式联系的猜想。

Derivations and identities for Fibonacci and Lucas polynomials

We introduce the notion of Fibonacci and Lucas derivations of the polynomial algebras and prove that any element of kernel of the derivations defines a polynomial identity for the Fibonacci and Lucas polynomials. Also, we prove that any polynomial identity for Appel polynomial yields a polynomial identity for the Fibonacci and Lucas polynomials and describe the corresponding intertwining maps.

Generalized Fibonacci and Lucas cubes arising from powers of paths and cycles

The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the independent sets of the h-th power of a path ordered by inclusion. For h=1 such a diagram is called a Fibonacci cube, and for h>1 we obtain a generalization of the Fibonacci cube. Consequently, we derive a generalized notion of Fibonacci sequence, called h-Fibonacci sequence. Then, we show that the number of edges of a generalized Fibonacci cube is obtained by con…

Binomial Fibonacci Power Sums

We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.

One Parameter Generalizations of the Fibonacci and Lucas Numbers

We give one parameter generalizations of the Fibonacci and Lucas numbers denoted by $\{F_n(þ)\}$ and $\{L_n(þ)\}$, respectively. We evaluate the Hankel determinants with entries $\{1/F_{j+k+1}(þ): 0\le i,j \le n\}$ and $\{1/L_{j+k+1}(þ): 0\le i,j\le n\}$. We also find the entries in the inverse of $\{1/F_{j+k+1}(þ): 0\le i,j \le n\}$ and show that all its entries are integers. Some of the identities satisfied by the Fibonacci and Lucas numbers are extended to more general numbers. All integer solutions to three diophantine equtions related to the Pell equation are also found.

Edges in Fibonacci cubes, Lucas cubes and complements

The Fibonacci cube of dimension n, denoted as $Γ\_n$, is the subgraph of the hypercube induced by vertices with no consecutive 1's. The irregularity of a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent paper based on the recursive structure of $Γ\_n$ it is proved that the irregularity of $Γ\_n$ and $Λ\_n$ are two times the number of edges of $Γ\_{n-1}$ and $2n$ times the number of vertices of $Γ\_{n-4}$, respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind (Figure 2) we give a bijective proof of both result…

斐波那契、卢卡斯、佩尔和雅各布斯塔尔塔的变种

The Towers of Fibonacci, Lucas, Pell, and Jacobsthal

本文介绍了四个新的汉诺塔变种问题，每个变种的最优解与斐波那契、卢卡斯、佩尔或雅各布斯塔尔数相关，并给出了每个变种的最优解及其关联图。

2×2矩阵对称幂的幂与迹以及组合、斐波那契和卢卡斯恒等式

Powers and trace of symmetric powers of $2\times 2$ matrices and combinatorial, Fibonacci and Lucas identities

研究2×2矩阵对称幂的幂与迹，通过计算二阶完全有向图上特定路径给出相关公式完整证明，与其他公式比较得出组合恒等式，尤其是斐波那契和卢卡斯数的表达式。

切比雪夫、斐波那契与卢卡斯多项式之间通过三角和建立的关系

Relations between Chebyshev, Fibonacci and Lucas polynomials via trigonometric sums

本文通过有限三角和建立斐波那契、卢卡斯与切比雪夫多项式间的新恒等式，并给出其物理解释及与图有效电阻的联系。

Areas of Triangles and other Polygons with Vertices from Various Sequences

Motivated by Elementary Problem B-1172 in the Fibonacci Quarterly (vol. 53, no. 3, pg. 273), formulas for the areas of triangles and other polygons having vertices with coordinates taken from various sequences of integers are obtained. The sequences discussed are Polygonal number sequences as well as Fibonacci, Lucas, Jacobsthal, Jacobsthal-Lucas, Pell, Pell-Lucas, and Generalized Fibonacci sequences. The polygons have vertices with the form $(p_n,p_{n+k}),\,\, (p_{n+2k},p_{n+3k}),\,\,\dots ,(p_{n+(2m-2)k},p_{n+(2m-1)k)}$.

轮图中的根生成森林：Fibonacci-Lucas 公式与极值根配置

Rooted spanning forests in wheel graphs: Fibonacci-Lucas formulas and extremal root configurations

本文研究轮图中的根生成森林，给出基于Fibonacci和Lucas数的计数公式，并解决根配置的极值问题。

Kostant's Weight Multiplicity Formula and the Fibonacci and Lucas Numbers

Consider the weight $λ$ which is the sum of all simple roots of a simple Lie algebra. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of the zero weight in the representation with highest weight $λ$. We prove that in Lie algebras of type $A$ and $B$, the number of contributing terms to the multiplicity of the zero-weight space in the representation with highest weight $λ$ is given by a Fibonacci number, and that in Lie algebras of type $C$ and $D$, the analogous result is given by a multiple of a Lucas number.

A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers

In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci- and Lucas numbers are obtained. Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci- and Lucas numbers are investigated.

斐波那契-富比尼数与卢卡斯-富比尼数

The Fibonacci-Fubini and Lucas-Fubini numbers

本文通过将有序贝尔数中的划分块按斐波那契方式排列，引入斐波那契-富比尼数，并研究其生成函数、显式公式、组合解释及算术三角形，同时揭示其与斯特林数的联系。

The (non-)existence of perfect codes in Lucas cubes

The Fibonacci cube of dimension n, denoted as $Γ$ n , is the subgraph of the n-cube 5 Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in $Γ$ n for n $\ge$ 4. As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes. The most direct generalization is the family $Γ$ n (1 s) of subgraphs induced by strings without 1 s as a substring where s $\ge$ 2 is a given integer. In a precedent work 10 we proved the existence of a perfect code in $Γ$ n (1 s) for n = 2 p -- 1 an…

On $k$-Fibonacci sums by matrix methods

In this paper, some $k$-Fibonacci and $k$-Lucas with arithmetic indexes sums are derived by using the matrices $R_{a}=\left[ \begin{array}{lr} L_{k,a} & -(-1)^{a} \\ 1 & 0 \end{array}\right]$ and $S_{a}=\frac{1}{2}\left[ \begin{array}{lr} L_{k,a} & Δ_{a} \\ 1 & L_{k,a} \end{array}\right]$, where $Δ_{a}=L_{k,a}^{2}-4(-1)^{a}$. The most notable side of this paper is our proof method, since all the identities used in the proofs of main theorems are proved previously by using the matrices $R_{a}$ and $S_{a}$, with $a$ an natural number. Although the identities we proved are known, our proofs are n…

On the properties of new families of generalized Fibonacci numbers

In this paper, new families of generalized Fibonacci and Lucas numbers are introduced. In addition, we present the recurrence relations and the generating functions of the new families for $k=2$.

二次代数中幂的通用恒等式及斐波那契恒等式的矩阵导出

Quadratic Reduction and Classical Multiple-Index Fibonacci-Lucas Identities

研究二次代数中元素幂的通用恒等式，通过矩阵迹和行列式推导斐波那契数列的二项式展开公式，揭示恒等式源于普遍代数原理。

Additional Fibonacci-Bernoulli relations

We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations of hyperbolic functions. Special cases and some corollaries will highlight interesting aspects of our findings.

广义斐波那契多项式的零点与正交性

Zeros and Orthogonality of generalized Fibonacci polynomials

本文研究了广义斐波那契多项式（GFP）的零点求解与正交性判定，给出了 Favard 定理特例的替代证明，分类了常见 GFP 的正交性，并探讨了正交多项式与 Markov 过程的联系。

GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sum…

斐波那契数和卢卡斯数来源于轮图两组件生成森林

Fibonacci and Lucas numbers arising from two-component spanning forests of wheel graphs

本文通过构造性双射和分析方法，推导出轮图两组件生成森林与斐波那契数、卢卡斯数的关系。

Balancing polynomials, Fibonacci numbers and some new series for $π$

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $π$ involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.

通过拟算术均值的间隙和及其在斐波那契和卢卡斯序列中的应用

Gap-Sums via Quasi-Arithmetic Means with Applications to Fibonacci and Lucas Sequences

本文通过拟算术均值统一研究递增整数序列中缺失整数的和，推广了Barry的算术间隙和到几何与调和情形，并应用于斐波那契和卢卡斯序列，得到调和间隙和指数收敛到ln(α)以及倒数和渐近展开式。

Elliptic solutions of dynamical Lucas sequences

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative ellipt…

Power law behavior associated with a Fibonacci Lucas model and generalized statistical models

A Fibonacci-Lucas based statistical model and several other related models are studied. The canonical and grand canonical partition functions for these models are developed.Partition structure such as the distribution of sizes as in a cluster distribution is explored.Ensemble averaging over all partitions leads to a scale invariant power law behavior at a particular critical like point. The canonical ensemble of the Fibonacci-Lucas case involves the Gegenbauer polynomial.The model has a hyperbolic power law behavior, a feature linked to the golden mean ratio of two adjacent Fibonacci numbers a…

关于 $ax+by+cz=n$ 解的个数：斐波那契与卢卡斯三元组

The Number of Solutions to $ax+by+cz=n$ for Fibonacci and Lucas triplets

本文研究了在斐波那契和卢卡斯三元组情况下，方程 $ax+by+cz=n$ 的非负整数解的个数，并推导出精确公式。

The $p$-Frobenius and $p$-Sylvester numbers for Fibonacci and Lucas triplets

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $a_1,a_2,\dots,a_l$ be positive integers such that their greatest common divisor is one. For a nonnegative integer $p$, denote the $p$-Frobenius number by $g_p(a_1,a_2,\dots,a_l)$, which is the largest integer that can be represented at most $p$ ways by a linear combination with nonnegative integer coefficients of $a_1,a_2,\dots,a_l$. When $p=0$, $0$-Frobenius number is the classical Frobenius number. When $l=2$, $p$-Frobenius number is explicitly given. However, when $l=3$ and even larger, even i…

关于Wythoff和、Fibonacci词与Lucas词的若干OEIS猜想的证明

Proofs of some OEIS conjectures on Wythoff sums, Fibonacci and Lucas words

本文用Walnut证明器证明了多个关于Wythoff序列、Fibonacci词和Lucas词的OEIS猜想，包括Kimberling的表示计数猜想与间隙猜想，并提供了完整命令文件。