A Matrix Dynamics Approach to Golomb's Recursion

Abstract

In an unpublished note Golomb proposed a family of "strange" recursions of metafibonacci type, parametrized by $k$. Previously we showed that contrary to Golomb's conjecture, for each $k$ there are many increasing solutions, and an explicit construction for multiple solutions was displayed. By reformulating our solution approach using matrix dynamics, we extend these results to a characterization of the asymptotic behaviour of all solutions of the Golomb recursion. This matrix dynamics perspective is also used to construct what we believe is the first example of a "nontrivial" nonincreasing solution, that is, one that is not eventually increasing.