Ömer Eğecioğlu, Elif Saygı, Zülfükar Saygı
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Daisy cubes are a class of isometric subgraphs of the hypercubes Q n. Daisy cubes include some previously well known families of graphs like Fibonacci cubes and Lucas cubes. Moreover they appear in chemical graph theory. Two distance invariants, Wiener and Mostar indices, have been introduced in the context of the mathematical chemistry. The Wiener index W (G) is the sum of distance between all unordered pairs of vertices of a graph G. The Mostar index Mo(G) is a measure of how far G is from being distance balanced. In this paper we establish that the Wiener and the Mostar indices of a daisy c…
Fibonacci, Lucas, and Spread Polynomials
本文基于Fibonacci和Lucas多项式，对spread多项式的一种变体进行了初等阐述。
In this paper, we discuss about some results on average of Fibonacci and Lucas sequences that may be found in the OEIS code A111035.
Identities for Fibonacci and Lucas polynomials derived from a book of Gould
这封信是献给古尔德教授的。其目的是显示如何身份在他的书“组合恒等式”可以用来获得身份的斐波那契和卢卡斯多项式。反过来，这些身份允许获得丰富的数字身份的斐波那契和卢卡斯数。
We present here some new identities for generalizations of Fibonacci and Lucas numbers by combinatorially interpreting these numbers in terms of numbers of certain tilings of a $1 \times m$ board. As a consequence, some new interesting identities involving the ordinaries Fibonacci and Lucas numbers are derived.
New Binomial Identities for Fibonacci, Lucas, and Generalized Fibonacci Sequences with Multiple Indices
本文提出新的恒等式，通过卢卡斯数的幂和二项式系数表示斐波那契、卢卡斯和广义斐波那契序列的多重索引项，利用对称多项式（瓦ring公式）应用于经典Binet公式。
The diameter of a graph is the maximum distance between pairs of vertices in the graph. A pair of vertices whose distance is equal to its diameter are called diametrically opposite vertices. The collection of shortest paths between diametrically opposite vertices are referred as diametral paths. In this work, we enumerate the number of diametral paths for Fibonacci cubes, Lucas cubes and Alternate Lucas cubes. We present bijective proofs that show that these numbers are related to alternating permutations and are enumerated by Euler numbers.
Some formulae for bivariate Fibonacci and Lucas polynomials
我们推导出一个集合的身份二元斐波那契和卢卡斯多项式使用基本上是一个矩阵的方法，以及这样的多项式的属性时，变量$x$和$y$被替换为多项式。对于选定的变量值，可以得到大量的组合恒等式。
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
证明了Fibonacci立方体和Lucas立方体的渐近平均偏心率和渐近平均度分别为$(5+\sqrt 5)/10$和$(5-\sqrt 5)/5$。引入了Fibonacci树叶子的一种新标号，并证明了给定Fibonacci立方体顶点的偏心率等于相应Fibonacci树叶子的深度。超立方体密度也被引入和研究。Fibonacci立方体和Lucas立方体的超立方体密度被证明是$(1-1/\sqrt 5)/\log_2φ$，其中$φ$是黄金比例，和图的笛卡儿积被用来构造具有固定的，非零的超立方体密度的图的家庭。证明了Fibonacci字和Lucas字的比值分别以固定坐标0和1的极限赋范和为$φ^2$。
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。
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
斐波那契立方体$Γ_n$是由不包含两个连续1的二进制串诱导的超立方体的子图。卢卡斯立方体$Λ_n$是从$Γ_n$通过移除以1开始和结束的顶点而获得的。我们刻画了$Γ_n$和$Λ_n$中的最大诱导超立方体，并对任意$p\leq n$导出了这些图中最大$p$维超立方体的个数.
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.
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
本文确定了Fibonacci八元数的一些性质。此外，我们引入了广义Fibonacci-Lucas八元数，并研究了这些元素的一些性质。
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
研究了Fibonacci多项式和Lucas多项式的$% \sum_{n=0}^{q}F_{tsn}^{k}(x) $和$\sum_{n=0}^{q}L_{tsn}^{k}% (x) $形式的幂和，其中$s,t,k$是给定的自然数，以及相应的交替和$\sum_{n=0}^{q}(-1) ^{n}F_{tsn}^{k}(x) $和$\sum_{n=0}^{q}(-1) ^{n}L_{tsn}^{k}(x) $ .给出了将这些和表示为某些$s$ -Fibopolynomial的线性组合的参数$s,t,k$的充分条件。
Routing linear permutations on Fibonacci and Lucas cubes
近年来，人们对超立方体的某些子立方体，即斐波那契立方体和卢卡斯立方体（及其广义版本）产生了很大的兴趣。在这篇文章中，我们考虑在线路由这些立方体上的线性排列。我们使用的路由模型将边视为双向的，并且不允许队列长度大于大于1。消息从不同的顶点开始，在与时钟同步的运动中，移动到相邻的顶点或保持在原地，这样在下一阶段，每个顶点仍然只有一条消息。这是我们在前面的文章中定义的路由模型。
Some beautiful q-analogues of Fibonacci and Lucas polynomials
我们给出了两类著名的基本性质的概述q-Fibonacci和q-Lucas多项式，并提供了一个共同的推广。
Based on well-known properties of Fibonacci and Lucas numbers and polynomials we give a self-contained approach to some bivariate analogs.
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。
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。
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.
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。
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
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。
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.
Sums of products of generalized Fibonacci and Lucas numbers
本文建立了广义Fibonacci数与Lucas数乘积的和与交错和的几个公式。特别地，我们恢复和推广了Z. Cerin和Z. Cerin & G. M.吉亚内拉，更容易。
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.
Vertex and edge orbits of Fibonacci and Lucas cubes
斐波那契立方体$Γ_n$是从$n$立方体$Q_n$中去掉所有包含两个连续1的顶点得到的。此外，如果删除以1开始和结束的顶点，则获得Lucas立方体$Λ_n$。在自同构群的作用下，确定了Fibonacci立方体和Lucas立方体的顶点轨道和边轨道的数目，轨道的大小集合，以及每种大小的轨道的数目.具体地，$Λ_n$的顶点轨道的大小的集合是$\{k \ge 1;\ k \divides n\} \cup\, \{k \ge 18;\ k \divides 2n\}$，大小为$k$的$Λ_n$的顶点轨道的数目等于$\sum_{d\divides k}μ\left(\frac{k}{d}\right) F_{\lfloor \frac{d}{2}\rfloor + 2}$，其中$k$是奇数并且整除$n$，并且$Λ_n$的边缘轨道的数目等于$Γ_{n-3}$的顶点轨道的数目。弦和本原弦的二面角变换是证明这些结果的必要工具。
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。
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
本文引入了多项式代数的Fibonacci和Lucas导子的概念，证明了导子核的任何元素都定义了Fibonacci和Lucas多项式的多项式恒等式。证明了Appel多项式的任何多项式恒等式都能得到Fibonacci多项式和Lucas多项式的多项式恒等式，并给出了相应的交织映射。
Generalized Fibonacci and Lucas cubes arising from powers of paths and cycles
本文讨论了Fibonacci序列和Lucas序列的一些推广，分别由路和圈的幂引起。在第一部分的工作中，我们提供了一个公式的边缘数的哈塞图的独立集的第h次幂的道路有序的列入。对于h=1，这样的图被称为斐波那契立方体，对于h>1，我们得到斐波那契立方体的推广。因此，我们得到了Fibonacci序列的一个广义概念，称为h-Fibonacci序列。然后，我们证明了广义Fibonacci立方体的边数是由h-Fibonacci序列与自身的卷积得到的。在第二部分中，我们考虑循环的情况。我们评估的边缘的数量的哈斯图的独立集的第h次幂的一个周期有序的包含。当h=1时，这样的图称为卢卡斯立方体，当h>1时，我们得到卢卡斯立方体的一个推广。然后，我们得到一个广义版本的卢卡斯序列，称为h-卢卡斯序列。最后，我们证明了广义Lucas立方体的边数是通过一个适当的卷积的h-Fibonacci序列的h-Lucas序列。
We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.
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…
One Parameter Generalizations of the Fibonacci and Lucas Numbers
我们给出了Fibonacci数和Lucas数的单参数推广，分别记为$\{F_n(þ)\}$和$\{L_n(þ)\}$.我们用条目$\{1/F_{j+k+1}(þ): 0\le i,j \le n\}$和$\{1/L_{j+k+1}(þ): 0\le i,j\le n\}$计算Hankel行列式。我们还找到了$\{1/F_{j+k+1}(þ): 0\le i,j \le n\}$的逆中的元素，并证明了它的所有元素都是整数。斐波那契数和卢卡斯数所满足的一些恒等式被推广到更一般的数。还找到了与Pell方程有关的三个丢番图方程的所有整数解。
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。
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。
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。
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 $(p_n,p_{n+k}),\,\, (p_{n+2k},p_{n+3k}),\,\,\dots ,(p_{n+(2m-2)k},p_{n+(2m-1)k)}$.
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。
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
在这项工作中，我们介绍了一个对称的算法得到的递归关系a_{n} ^{k}= a_{n-1} ^{k}+ a_{n} ^{k-1}。指出该算法可应用于超调和数、普通数和不完全Fibonacci数及Lucas数。给出了Fibonacci数和Lucas数的超调和数和一般生成函数的显式表达式。此外，我们还定义了“超斐波纳契数”、“超卢卡斯数”。利用这些新的概念，普通和不完全Fibonacci-和Lucas数之间的一些关系进行了研究。
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…
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。
Some new results about Fibonacci p-cubes
本文针对斐波那契p-立方体$Γ_n^p$，证明了其立方体多项式的猜想表达式，用新方法确定距离立方体多项式，并补充了Mostar指数和不规则度两个新不变量。
On $k$-Fibonacci sums by matrix methods
本文利用矩阵$R_{a}=\left[ \begin{array}{lr} L_{k,a} & -(-1)^{a} \\ 1 & 0 \end{array}\right]$和$S_{a}=\frac{1}{2}\left[ \begin{array}{lr} L_{k,a} & Δ_{a} \\ 1 & L_{k,a} \end{array}\right]$，导出了一些算术指标和的$k$ -Fibonacci和$k$ -Lucas，其中$Δ_{a}=L_{k,a}^{2}-4(-1)^{a}$ .本文最值得注意的方面是我们的证明方法，因为在证明主要定理中使用的所有恒等式都是使用矩阵$R_{a}$和$S_{a}$证明的，其中$a$是自然数。虽然我们证明的身份是已知的，我们的证明是没有遇到的$k$ -Fibonacci和$k$ -卢卡斯数文献。
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 $k=2$.
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。
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.
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。
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。
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…
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.
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。
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
研究了一个基于Fibonacci-Lucas的统计模型和几个相关的模型。给出了这些模型的正则和巨正则配分函数，探讨了诸如簇分布中的尺寸分布等配分结构，在所有配分上的包络平均导致了在特定的类临界点处的标度不变幂律行为. Fibonacci-Lucas情形的正则系综涉及Gegenbauer多项式，该模型具有双曲幂律行为，这一特征与两个相邻Fibonacci数的黄金均值比以及Lucas数与黄金均值的关系有关。与其他幂律行为的关系，如齐普夫和帕累托定律，被提及。对于所考虑的情况，巨正则系综涉及高斯超几何函数F（a，b，c，z），其中a，b，c具有特定值。一般情况下具有τ指数等于1 +c-a-b的可变幂律行为。a=1/2，b=1，c=3，因此τ =5/2的应用非常接近玻色-爱因斯坦凝聚。精确理论的zeta函数zeta（3/2）=2.61被替换为8/3，zeta（5/2）= 1.34被替换为4/3。在凝聚点，长度的圈数下降为标度不变的幂律。这些循环来自与玻色-爱因斯坦统计相关的排列对称性，可以被视为粒子之间复杂网络的链接。这种尺度不变的幂律与复杂网络中的幂律相似。网络的增长是使用模型的递归特性开发的。规范系综和相关性所施加的约束导致Fibonacci和Lucas数之间的一些数论联系，作为这种方法的附带结果。
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。