MathNet 2002 — Part 02 Question 89

A sequence $s_{0}, s_{1}, s_{2}, s_{3}, \ldots$ is defined by $s_{0}=s_{1}=1$ and, for every positive integer $n$, $s_{2 n}=s_{n}$, $s_{4 n+1}=s_{2 n+1}$, $s_{4 n-1}=s_{2 n-1}+s_{2 n-1}^{2} / s_{n-1}$. What is the value of $s_{1000}$?