论文标题
涉及通过梳子平方的Tribonacci数字的身份
Identities involving the tribonacci numbers squared via tiling with combs
论文作者
论文摘要
用$（\ frac12，\ frac12; 1）$（\ frac12，\ frac12，\ frac12; 2; 2; 2）$ - 和$（\ frac12，\ frac12，\ frac12; 3）$ combs $ t $ comb $ combs $ t y $ t y $ t y $ t y $ t_ 2 n $ n wery $ comb $ t y $ t there $ n wery $ comb $ te_ 2 n wery $ comb $ t y ther $ n $ tribonacci编号。 a $（\ frac12，\ frac12; m）$ - 梳子是一个瓷砖，由$ m $ subs tiles组成，尺寸为$ \ frac12 \ times1 $（较短的侧面总是水平的），由尺寸$ \ frac12 \ times1 $的尺寸间隔隔开。我们使用这样的瓷砖来获取与彼此之间的部落数字相关的三个群体数字，与部落数字的其他组合以及斐波那契，纳拉亚娜的母牛和帕多瓦人数字有关的快速组合证明。这些身份大多数似乎都是新的。
The number of ways to tile an $n$-board (an $n\times1$ rectangular board) with $(\frac12,\frac12;1)$-, $(\frac12,\frac12;2)$-, and $(\frac12,\frac12;3)$-combs is $T_{n+2}^2$ where $T_n$ is the $n$th tribonacci number. A $(\frac12,\frac12;m)$-comb is a tile composed of $m$ sub-tiles of dimensions $\frac12\times1$ (with the shorter sides always horizontal) separated by gaps of dimensions $\frac12\times1$. We use such tilings to obtain quick combinatorial proofs of identities relating the tribonacci numbers squared to one another, to other combinations of tribonacci numbers, and to the Fibonacci, Narayana's cows, and Padovan numbers. Most of these identities appear to be new.
