与帕斯卡三角和斐波那契数列相关的三对角实对称矩阵

Fibonacci q-gaussian sequences

The summation formula within pascalian triangle resulting in the fibonacci sequence is extended to the $q$-binomial coefficients $q$-gaussian triangles.

新的二项式恒等式用于斐波那契、卢卡斯和广义斐波那契序列的多重索引

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。

New presences of $π$ and $e$ in Pascal's triangle

The following work shows new connections between the constants $π$ and $e$ with Pascal's triangle and the Lucas triangle, established via Fibonacci polynomials and similar means. Furthermore, relations between the two famous constants and the rows of Pascal's triangle and the Lucas triangle are conjectured, together with some other important related identities.

Pascal 三角形中的 Pascal 平铺与模 N 同余

Pascal tiling and congruences modulo N in Pascal's triangle

本文研究 Pascal 三角形前 N 行经几何变换所得矩阵的同余性质，证明其构成 Pascal 平铺当且仅当 N 为素数，并由此给出 Lucas 数同余式的替代证明，同时探讨 Fibonacci 伪素数情形及对多项式系数的推广。

Counting the number of non-zero coefficients in rows of generalized Pascal triangles

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\ge 0}$ counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for $(S(n))_{n\ge 0}$. This leads to a connection with the $2$-regular Stern-Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on…

On Two Families of Generalizations of Pascal's Triangle

We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by $m-1$ zeros on the right side. The $m=1$ cases are Pascal's triangle and the two families also coincide when $m=2$. Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the $m$-th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the $(1/(1-x^m),x/(1-x))$ Riordan array. Properties of this family of triangles can be obtained quickly as a result. The $(n,k)$-th entry in the $m$-th member…

A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence

The Stern polynomials defined by $s(0;x)=0$, $s(1;x)=1$, and for $n\geq 1$ by $s(2n;x)=s(n;x^2)$ and $s(2n+1;x)=x\,s(n;x^2)+s(n+1;x^2)$ have only 0 and 1 as coefficients. We construct an infinite lower-triangular matrix related to the coefficients of the $s(n;x)$ and show that its inverse has only 0, 1, and $-1$ as entries, which we find explicitly. In particular, the sign distribution of the entries is determined by the Prouhet-Thue-Morse sequence. We also obtain other properties of this matrix and a related Pascal-type matrix that involve the Catalan, Stirling, Fibonacci, Fine, and Padovan n…

On a Two-Parameter Family of Generalizations of Pascal's Triangle

We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer $N$) with unit squares and $(1,m-1;t)$-combs for some fixed $m=1,2,\dots$ and $t=2,3,\dots$ that use $n$ tiles in total of which $k$ are combs. A $(1,m-1;t)$-comb is a tile composed of $t$ unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width $m-1$. We show that the entries in the triangle are co…

关于Gullerud、Johnson和Mbirika猜想的注记

A Note on Conjectures of Gullerud, Johnson, and Mbirika

本文探讨Gullerud、Johnson和Mbirika关于三对角实对称矩阵根分布的猜想，部分验证了两个猜想，发现一类多项式不满足第三个猜想，并质疑第四个猜想。