On a Generalized Fibonacci Recurrence

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].