2019/01/13 by Natasha Blitvić, Blitvić, Natasha, Vicente I. Fernandez +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Combinatorics (math.CO) #Diffusion and Search Dynamics #FOS: Biological sciences #FOS: Mathematics #Fractal and DNA sequence analysis #Populations and Evolution (q-bio.PE) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1901.04080
openalex publication_date 2019/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The generalized Fibonacci recurrence gn=gn-k+gn-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 N0× \mathbb N0\ni(x,y)↦ x+y\choose x, previously sought by Dickinson [Dic50], Raab [Raa63], and Green [Gre68].