Mario Catalani
We consider sequences of the type $A_n=6A_{n-1}-A_{n-2}, A_0=r, A_1=s$ ($r$ and $s$ integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four particular sequences of the same type.
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
Pell and associated Pell braid sequences as GCDs of sums of $k$ consecutive Pell, balancing, and related numbers
研究了六个著名序列（Pell、相伴Pell、平衡及相关数）中$k$个连续项之和的最大公约数，给出了依赖$k$的闭式，并发现其可表示为Pell和相伴Pell辫序列，同时探讨了平方项之和的GCD问题。
On the period of Pell-Narayana sequence in some groups
本文研究模m的Pell-Narayana序列，定义2-生成群的Pell-Narayana轨道，考察其周期长度，并给出多面体群及若干群（如Q8及其扩张）的周期结果。
In this paper, we investigate the generalized Pell sequence, the generalized complex Pell sequence and the generalized dual Pell sequence using the Pell numbers. We obtain special cases of these sequences. Furthermore, we give recurrence relations, vectors, the silver ratio and Binet's formula for the above-investigated sequences.
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。
Pell-Padovan tetranacci numbers and their Hadamard product with classical sequences
本文提出了一种快速计算佩尔-帕多瓦四阶斐波那契数与经典序列生成函数的方法，并能同时处理八个特殊实例。
In this paper some properties of generalized tribonacci and generalized Padovan sequence are presented. Also the Euclidean norms of circulant, $r$-circulant, semi-circulant and Hankle matrices with above mentioned sequences are calculated. The upper and lower bounds of spectral norms are also obtained.
On the Algebraic Properties of r-circulant Matrices Associated with Generalized k-Pell-Tribonacci Numbers
研究r-循环矩阵的代数性质，推导Frobenius范数和ℓ₁范数的显式表达式，给出特征值和行列式的闭式公式，并建立谱范数的上下界，推广了已有结果并提供更精确的界。
In this paper we extend the notion of Melham sum to the Pell and Pell-Lucas sequences. While the proofs of general statements rely on the binomial theorem, we prove some spacial cases by the known Pell identities. We also give extensions of obtained expressions to the other recursive sequences.
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。
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.
Let $a,b,c $ be any positive integers such that $c\mid ab$ and $d_i^\pm$ is a square free positive integer of the form $d_i^\pm=a^{2k} b^{2l}\pm i c^m$ where $k,l \geq m$ and $i=1,2.$ The main focus of this paper to find the fundamental solution of the equation $ x^2-d_i^\pm y^2=1$ with the help of the continued fraction of $\sqrt{d_i^\pm}.$ We also obtain all the positive solutions of the equations $ x^2-d_i^\pm y^2=\pm 1$ and $ x^2-d_i^\pm y^2=\pm 4$ by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form $ F_{d_i^\pm}(x, …
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 $(p_n,p_{n+k}),\,\, (p_{n+2k},p_{n+3k}),\,\,\dots ,(p_{n+(2m-2)k},p_{n+(2m-1)k)}$.
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。
Polynomial Identities for Generalized Fibonacci Sequences
该研究针对斐波那契、卢卡斯等序列的多项式恒等式，补充了广义斐波那契序列的相关恒等式，通过吉拉尔-沃林公式、比内公式结合线性代数两种方法完成序列分类，填补了数学文献中该类恒等式通用表述的空白。
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.
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular sequence as a unification of several kinds of important sequences including $k$-regular, $k$-additive and $k$-multiplicative sequences. In this paper, we give a generalization of the result of Bell, Bugeaud and Coons for certain generalized $k$-regular series. Especially, we show that the values of irrational generating functions of certain sum of $k$-additiv…
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。
Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and Polychronakos in 2019. We give a bijection between these two varieties of compositions and determine various combinatorial properties of these multicompositions. In particular, we determine the count of multicompositions by number of all parts, number of positive parts, and number of zeros. Then, working from three types of compositions with restricted parts that are counted by the Fibonacci seque…
Motivated by the known autocorrelation properties of the Rudin-Shapiro sequence, we study the discrete correlation among infinite sequences over a finite alphabet, where we just take into account whether two symbols are identical. We show by combinatorial means that sequences cannot be "too" different, and by an explicit construction generalizing the Rudin-Shapiro sequence, we show that we can achieve the maximum possible difference.
We determine the degree sequence of the generalized Sierpinski graph and its general first Zagreb index in terms of the same parameters of the base graph G.
Existence of Solutions of Functional-Difference Equations with Proportional Delay on Deformed Generalized Fibonacci Polynomials via Successive Approximation and Bell Polynomials
本文构建变形广义 Fibonacci 多项式上的微积分为框架，借助变形指数函数、逐次逼近法和 Bell 多项式证明比例延迟泛函差分方程解的存在性，并指出 q-周期情形下解不唯一。
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between integer partitions with kinds and finite differences of the coefficients of generalized Galois numbers is established.
Conditions for building generalized action graphs from sequences
该研究聚焦与数序列紧密相关的广义作用图，从结构与组合角度探究序列生成这类图的条件，基于前期研究分析图特征与序列模式的对应关系，构建了识别可生成广义作用图的序列的框架。
A Novel Property of Generalized Fibonacci Sequence in Grids
本文研究了奇数阶网格中主对角线与副对角线上数字和的比值，揭示其仅依赖于网格阶数，提出一个简洁优雅的恒等式。
In this note we use the analogy between the Catalan sequence and the Rueppel sequence to derive a variety of conjectures surrounding the Hankel transforms of a number of sequences closely related to the Rueppel sequence. Use is made of the representation of suitable generating functions by Stieltjes continued fractions. We define polynomial sequences by introducing parameters that define generalized Rueppel sequences, and we show that such polynomials have coefficient arrays that are Riordan arrays. Finally we conjecture the form of a product of Hankel transforms arising from the Rueppel seque…
The Tribonacci sequence is a well-known example of third order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Tribonacci sequence is introduced and defined by $H_{n+2}=H_{n+1}+H_{n}+H_{n-1}\ \ (n\geq 1)$, where $H_{0}=3$, $H_{1}=0$ and $H_{2}=2$. Also $n$-th power of the generating matrix for this generalized Tribonacci sequence is established and some basic properties of this sequence are obtained by matrix methods. There are many elementary formulae relating the various $H_{n}$, most of which, since the sequence is defined …
Hurwitz Stability of Generalized Turán Expressions of Polynomial Sequences
本文证明了Eulerian和Narayana多项式的Turán表达式稳定性猜想，推广了Turán表达式概念，给出广义稳定性判定准则，并应用于递推多项式序列。
The Weighted Tower of Hanoi: Algebraic Structure, Phase Transitions, and Integer Sequences
本文研究带权重的汉诺塔问题，通过代数理论分析其相变现象，并揭示其与整数序列的深刻联系。
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。