A simple proof for generalized Fibonacci numbers with dying rabbits

Abstract

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 $d$.