Arithmetic properties of q-Fibonacci numbers and q-Pell numbers

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…

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 $(p_n,p_{n+k}),\,\, (p_{n+2k},p_{n+3k}),\,\,\dots ,(p_{n+(2m-2)k},p_{n+(2m-1)k)}$.

广义佩尔图

Generalized Pell graphs

本文引入广义佩尔图，研究其度量、计数与结构性质，确定边数和立方多项式的生成函数，描述中心并证明其是中位图且可嵌入斐波那契立方体。

Generalized Compositions of Natural Numbers

We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known classes of integers such as Fibonacci, Catalan, Pell, Pell-Lucas, and Jacobsthal numbers.

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。

佩尔塔与奥斯特罗数论

The Pell Tower and Ostronometry

研究由递推关系 X_{n+1}=dX_n+X_{n-1} 定义的表格，探索新模式并遇到红墙和奇异计数系统。

Pell序列及相关序列的连续项之和

Sum of Consecutive Terms of Pell and Related Sequences

该研究推导Pell序列连续项和的恒等式，明确N个连续Pell数和为另一Pell数固定整数倍的充要条件，推广至广义Pell(k,i)数及Fibonacci等二阶递归序列并给出对应结论。

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.

Hurwitzian continued fractions containing a repeated constant and an arithmetic progression

We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials. The limit formula is a generalization of Lehmer's theorem that implies the continuous fraction expansions of $e$ and $\tan(1)$, and it can also be derived from Lehmer's work using Fibonacci polynomial i…

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。

Fibonacci、Thue-Morse 和 Rudin-Shapiro 词中的单色等差数列

Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words

本文研究 Fibonacci 词中固定公差最长单色等差数列的长度与位置，结合圆周旋转动力学与 Walnut 自动定理证明给出长度公式，并推广到 Thue-Morse 和 Rudin-Shapiro 词。

Partitions enumerated by self-similar sequences

The Fibonacci numbers are the prototypical example of a recursive sequence, but grow too quickly to enumerate sets of integer partitions. The same is true for the other classical sequences $a(n)$ defined by Fibonacci-like recursions: the tribonacci, Padovan, Pell, Narayana's cows, and Lucas sequences. For each sequence $a(n)$, however, we can define a related sequence $\textrm{sa}(n)$ by defining $\textrm{sa}(n)$ to have the same recurrence and initial conditions as $a(n)$, except that $\textrm{sa}(2n)=\textrm{sa}(n)$. Growth is no longer a problem: for each $n$ we construct recursively a set …

Fibonacci自动序列的线性子序列复杂性

Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

研究Fibonacci自动序列的线性子序列的复杂性，分析自动机状态数及基本操作的复杂性，改进Bosma和Don的界。

Lucas序列中的周期模式、进入点与阶：理论与应用

Period patterns, entry points, and orders in the Lucas sequences: theory and applications

本文将模 m 的 Fibonacci 和 Lucas 序列统计理论扩展至 Lucas 序列 U 和 V，基于阶、周期与进入点等概念推广了统计恒等式，并应用于新颖的图解方法以揭示序列项的周期模式。

金属立方体

Metallic cubes

本文研究推广斐波那契立方体和佩尔图的双参数图族，证明其为超立方体的诱导子图并给出规范分解，计算度量不变量和度分布，建立哈密顿性质，为潜在应用奠定基础。

Pell 三元组的 $p$-数值半群

$p$-numerical semigroups of Pell triples

本文针对基于 Pell 型数递推关系构建的 Pell 数值半群，利用 $p$-Apéry 集，通过逐案讨论推导出了其 $p$-Frobenius 数和 $p$-亏格的显式公式。

蛇图中的组合联系：铺砌、格路径与完美匹配

Combinatorial connections in snake graphs: Tilings, lattice paths, and perfect matchings

本文引入三角蛇图，建立其路径、底层蛇图完美匹配与铺砌之间的双射，并据此给出直蛇图完美匹配数的 Hankel 行列式表达及 Fibonacci、Pell 序列的行列式来源。

Applications of some special numbers obtained from a difference equation of degree three

In this paper we present applications of some special numbers obtained from a difference equation of degree three, especially in the Coding Theory. As a particular case, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternions.

针对斐波那契数及其他序列的改进拉姆齐型定理

Improved Ramsey-type theorems for Fibonacci numbers and other sequences

该研究聚焦范德瓦尔登数的两类变体，以斐波那契数集为公差集合开展分析，推导得出两类拉姆齐型数存在性的改进上界，同时补充了其他集合的相关计算数据。

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let $\UT_n(\FF_q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of $\UT_{n+1}(\FF_q)$ are indexed by lattice paths from the origin to the line $x+y=n$ using the steps $(1,0), (1,1), (0,1), (1,1)$, which are labeled in a certain way by nonzero elements of $\FF_q$. In particular, we prove for $n\geq 1$ that the number of Heisenberg characters of $\UT_{n+1}(\FF_q)$ is a polynomial in $q-1$ with nonnegative integer coefficients and degree $n$, whose leading coefficient is the $n$th Fibonacci number…

Farey 分形中出现的一些结构

Some Structures arising from the Farey Fractal

研究Farey型图、斐波那契展开与操作构造的相互作用，引入定义为Zeckendorf移位下的逆极限的Fibonadic数，配备度量、顺序和交换体结构。

关于斐波那契密度上无限词 $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ 的改进遍历理论

Improvement Ergodic Theory For The Infinite Word $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ on Fibonacci Density

本文研究斐波那契密度上无限词的遍历理论，分析其子词多样性与广义模式分布，并揭示替换规则、代数数论与组合复杂性之间的深层联系。

Polynomials and Second Order Linear Recurrences

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery, Hilbert's question whether there exists a general algorithm for checking the solvability in integers of any diophantine equation. In this paper we describe techniques to prove the nonexistence of polynomials in two variables for some simple generalizations of the Fibonacci sequence (explicit diophantine representation of Fibonacci numbers were known from Jones…

Asymptotic Normality of the Coefficients of the Morgan-Voyce Polynomials

We study arithmetic and asymptotic properties of polynomials provided by $Q_n(x):= x \sum_{k=1}^n k \, Q_{n-k}(x)$ with initial value $Q_0(x)=1$. The coefficients satisfy a central limit theorem and a local limit theorem involving Fibonacci numbers. We apply methods of Berry and Esseen, Harper, Bender, and Canfield.

旋转记忆斐波那契数与周期平铺

Rotating-Memory Fibonacci Numbers and Periodic Tilings

本文提出旋转记忆斐波那契数，一种递推记忆周期性变化的数列，证明其刚性结构、闭式与生成函数，并通过周期平铺给出几何解释，为更高周期建模提出开放问题。