The dying rabbit problem revisited

TL;DR

This paper generalizes the Fibonacci sequence by incorporating rabbit mortality and maturation delay, providing new recurrence relations, explicit formulas, and analyzing the roots of associated polynomials.

Contribution

It introduces a generalized recurrence relation for the sequence, extending previous models, and explicitly computes the sequence's general term.

Findings

Derived a new recurrence relation for the generalized sequence.

Explicitly calculated the general term of the sequence.

Analyzed the behavior of the roots of the characteristic polynomial.

Abstract

In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in the paper Hoggatt, V. E., Jr.; Lind, D. A. "The dying rabbit problem". Fibonacci Quart. 7 1969 no. 5, 482--487) and we calculate explicitly their general term (extending the work in the paper Miles, E. P., Jr. Generalized Fibonacci numbers and associated matrices. Amer. Math. Monthly 67 1960 745--752). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences.