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$k$连续Fibonacci、Lucas和广义斐波那契数和的GCD

GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

Dan Guyer, aBa Mbirika

备注：
25 pgs. This version of the paper is almost identical to the published version. This arXiv version differs from the published version in two respects: (1) the arXiv version contains a Table of Contents, and (2) the J. Integer Seq. version does not number theorems/lemmas/propositions/etc. by section, whereas this arXiv version does. Those two changes may make this arXiv version easier to navigate
发表信息：
Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.8
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摘要：

本文研究了广义Fibonacci数列$\left(G_n\right)_{n \geq 0}$中$k$连续项的和，该数列由递推式$G_n = G_{n-1} + G_{n-2}$给出，对于所有具有整数初始条件$G_0$和$G_1$的$n \geq 2$.特别是，我们给出了精确的值的最大公约数（GCD）的所有总和的$k$连续条款的$\left(G_n\right)_{n \geq 0}$。当$G_0 = 0$和$G_1 = 1$时，我们得到所有$k$连续斐波那契数和的GCD，当$G_0 = 2$和$G_1 = 1$时，我们得到所有$k$连续卢卡斯数和的GCD。用符号$\mathcal{G}_{G_0, G_1}\!(k)$表示$k$连续广义斐波那契数的所有和的GCD，我们给出了这些值的两个诱人的特征，一个涉及$k$中的一个简单公式，另一个涉及广义Pisano周期：$$\mathcal{G}_{G_0, G_1}\!(k) = \gcd(G_{k+1}-G_1,\, G_{k+2}-G_2)\; \mbox{and}$$ $$\mathcal{G}_{G_0, G_1}\!(k) = \mathrm{lcm}\{m \mid π_{G_0,G_1}\!(m) \text{ divides } k\},$$其中$π_{G_0,G_1}\!(m)$表示模$m$的广义斐波那契序列的广义Pisano周期。这些看起来截然不同的公式重合的事实导致了对斐波那契数和卢卡斯数的一些令人惊讶和令人愉快的新理解。

英文摘要：

We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sums of $k$ consecutive generalized Fibonacci numbers by the symbol $\mathcal{G}_{G_0, G_1}\!(k)$, we give two tantalizing characterizations for these values, one involving a simple formula in $k$ and another involving generalized Pisano periods: $$\mathcal{G}_{G_0, G_1}\!(k) = \gcd(G_{k+1}-G_1,\, G_{k+2}-G_2)\; \mbox{and}$$ $$\mathcal{G}_{G_0, G_1}\!(k) = \mathrm{lcm}\{m \mid π_{G_0,G_1}\!(m) \text{ divides } k\},$$ where $π_{G_0,G_1}\!(m)$ denotes the generalized Pisano period of the generalized Fibonacci sequence modulo $m$. The fact that these vastly different-looking formulas coincide leads to some surprising and delightful new understandings of the Fibonacci and Lucas 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。

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