A simple proof for generalized Fibonacci numbers with dying rabbits

TL;DR

This paper studies a generalized Fibonacci problem where rabbits mature after age $f$ and die after age $d$, and derives a simple, counting-based recurrence for $F_n$ expressed in terms of at most three preceding terms. By introducing age-structured counts $F_n^x$ and a base equation $F_n = F_{n-1} + newborns_n - deaths_n$, the authors unravel the dynamics through four regimes (Case 1–Case 4) and obtain a piecewise recurrence that collapses to $F_n = F_{n-1} + F_{n-f} - F_{n-d-1}$ for $n \ge d+2$, with explicit edge cases. This formulation generalizes known sequences: Fibonacci ($f=2,d=\infty$), Padovan ($f=2,d=3$), Tribonacci, and Tetranacci, among others, all arising as specific instantiations. The approach provides a transparent, purely combinatorial justification for the recurrence, contrasting with generating-function methods in prior work and offering a concise framework for analyzing aging and death in population recurrences. Overall, the paper advances understanding of generalized Fibonacci dynamics and presents a practical, accessible proof technique grounded in counting arguments.

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 $d$.

Theorems & Definitions (5)

• Definition 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Theorem 4.5