Proof Writing & Problem Solving · Superchallenge problem

Define $s(0) = 0$, $s(1) = 1$, and for $n \ge 1$: $s(2n) = s(n)$ and $s(2n + 1) = s(n) + s(n + 1)$. Find $s(512) + s(513) + s(514) + \cdots + s(1023)$.