On a strange recursion of Golomb (Q1909975)

\textit{S. W. Golomb} [Discrete chaos: sequences satisfying ``strange'' recursions, preprint (undated, likely late eighties or early nineties)] proposed a family of ``strange'' recursions of metafibonacci type, parametrized 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.