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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 $z_1 \in \mathbbR_{> 0},\ z_2 \in \mathbb R_{> 0}$. As a corollary, we derive an alternateproof of a recurrence for an expression involving double factorials which firstappeared in the 2004 Putnam Contest. We subsequently pose several openquestions suitable for motivated undergraduate students.

A recurrence for an expression involving double factorials