On a Strange Recursion of Golomb

Abstract

Golomb proposed a family of "strange" recursions of metafibonacci type, parameterized by $k$, and, for each $k$, identified what he speculated was the unique increasing solution. We show that, to the contrary, there are many increasing solutions for each $k$, and we indicate explicitly how to construct them. We also provide some additional general results concerning the nature of the strictly increasing solutions for this unusual family of recursions.