# On a Generalized Fibonacci Recurrence

## TL;DR

This paper explores a generalized Fibonacci recurrence as a model for asymmetric branching, providing a diagrammatic representation and an explicit binomial identity related to Pascal's triangle, with applications in biological aging.

## Contribution

It introduces a new abstract perspective on the generalized Fibonacci recurrence, linking it to branching processes and deriving a novel binomial identity for diagonal sums in Pascal's triangle.

## Key findings

- Derived a compact diagrammatic representation of the recurrence.
- Established an explicit binomial identity for sums along diagonals in Pascal's triangle.
- Connected the recurrence to models of asymmetric branching and biological aging.

## Abstract

The generalized Fibonacci recurrence $g_n=g_{n-k}+g_{n-m}$ was recently used to demonstrate the theoretically optimal nature of limited senescence in morphologically symmetrically dividing bacteria. Here, we study this recurrence from a more abstract viewpoint, as a general model for asymmetric branching, and interpret solutions for different initial conditions in terms of branching-related quantities. We provide a compact diagrammatic representation for the evolution of this process which leads to an explicit binomial identity for the sums of elements lying on the diagonals $kx+my=n$ in Pascal's triangle $\mathbb N_0\times \mathbb N_0\ni(x,y)\mapsto {x+y\choose x}$, previously sought by Dickinson [Dic50], Raab [Raa63], and Green [Gre68].