标题： 斐波那契数列的一般化证明：消亡兔子问题 显示英文标题

标题： A simple proof for generalized Fibonacci numbers with dying rabbits

摘要： 我们考虑了一般化的斐波那契计数问题，其中兔子在年龄为$f$时变得具有繁殖能力，并在年龄为$d$时死亡，$1<=f<=d$和$d$可以是有限或无限的。 我们提供了一个简单的证明，该证明完全基于计数论证，得到了一个递归公式，该公式可以将第$n$个一般化斐波那契数表示为最多前 3 个数的函数。 该公式概括了原始的斐波那契序列（对于$f=2$和$d=\infty$，或者$f=1$和$d=2$）以及其他与斐波那契相关的序列，如对于$f=2$和$d=3$的帕多瓦序列，对于$f=1$和$d=3$的三阶斐波那契数列，对于$f=1$和$d=4$的四阶斐波那契数列，以及对于$f=1$和有限值$d$的类似序列。 显示英文摘要

摘要： We consider the generalized Fibonacci counting problem with rabbits that become fertile at age $f$ and die at age $d$, with $1<=f<=d$ and $d$ finite or infinite. We provide a simple proof, based exclusively on a counting argumentation, for a recursive formula that gives the $n$th generalized Fibonacci number as a function of at most 3 previous numbers. The formula generalizes both the original Fibonacci sequence, for $f=2$ and $d=\infty$ (or $f=1$ and $d=2$), and other Fibonacci-related sequences, such as the Padovan sequence, for $f=2$ and $d=3$, the Tribonacci, for $f=1$ and $d=3$, Tetranacci, for $f=1$ and $d=4$, and alike sequences, for $f=1$ and finite values of $d$.