A recurrence for an expression involving double factorials

TL;DR

This paper derives a closed-form solution for a recurrence relation involving positive real numbers and uses it to provide an alternative proof for a double factorial recurrence from the 2004 Putnam Contest, also proposing open questions.

Contribution

It introduces a new closed-form solution for a specific recurrence and offers an alternative proof for a known double factorial recurrence, expanding understanding of these mathematical sequences.

Findings

Closed-form solution for the recurrence relation.

Alternative proof for a double factorial recurrence.

Open questions for further research.

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.