HMMT 2002 U29

Guts
Answer

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 $s_{1000}$ ?

Dataset text NuminaMath olympiads. Quoted with source.

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