两类广义斐波那契数平方与(0,1) Toeplitz矩阵积和式之间的联系
Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices
AI总结：
本文通过引入梳子瓷砖的铺砌方法，给出广义斐波那契数乘积的组合解释，并据此证明其平方与特定(0,1) Toeplitz矩阵积和式之间的卷积型恒等式。
摘要：
通过考虑用我们称为梳子的新型瓷砖铺砌N-板（由N个单位宽正方形单元组成的线性阵列），我们给出了两个连续广义斐波那契数s_n（其中s_n=Σ_{i=1}^q v_i s_{n-m_i}，s_0=1，s_{n<0}=0，v_i和m_i为正整数且m_1<...<m_q）各自取任意非负整数次幂后乘积的组合解释。一个(w,g;m)-梳子是由m个尺寸为w×1的矩形子瓷砖组成、中间由宽度为g的间隙分隔的瓷砖。该解释被用于给出新的卷积型恒等式的组合证明，这些恒等式将q=2、v_i=1、m_1=M、m_2=m+1（其中M=0,m）情形下的s_n^2与具有3条非零对角线的(0,1) Toeplitz矩阵的积和式联系起来，这3条对角线分别位于主对角线上方-2、M-1和m处。当m=1时，这些恒等式简化为联系Padovan数与Narayana奶牛数的恒等式。
英文摘要：
By considering the tiling of an $N$-board (a linear array of $N$ square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers $s_n$ (where $s_{n}=\sum_{i=1}^q v_i s_{n-m_i}$, $s_0=1$, $s_{n<0}=0$, where $v_i$ and $m_i$ are positive integers and $m_1<\cdots<m_q$) each raised to an arbitrary non-negative integer power. A $(w,g;m)$-comb is a tile composed of $m$ rectangular sub-tiles of dimensions $w\times1$ separated by gaps of width $g$. The interpretation is used to give combinatorial proof of new convolution-type identities relating $s_n^2$ for the cases $q=2$, $v_i=1$, $m_1=M$, $m_2=m+1$ for $M=0,m$ to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are $-2$, $M-1$, and $m$ above the leading diagonal. When $m=1$ these identities reduce to ones connecting the Padovan and Narayana's cows numbers.
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。
