A recurrence for an expression involving double factorials

Abstract

We find a closed-form solution to the recurrence $z_{n+2}=\frac{1}{z_{n+1}}+z_n$, where $n\in {\mathbb{Z}}_{\ge 1}$ and $z_1\in {\mathbb{R}}_{>0},z_2\in {\mathbb{R}}_{>0}$. As a corollary, we derive an alternate proof of a recurrence for an expression involving double factorials which first appeared in the 2004 Putnam Contest. We subsequently pose several open questions suitable for motivated undergraduate students.