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 $n \in \mathbb Z_{\geq 1}$ 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.