Let $s_n$ be the $n$-th positive integer that does not contain three consecutive ones in its binary representation.
For example, $s_1=1$ and $s_7=8$.

Define $F(N)$ to be the sum of $n^2$ for all $n\le N$ where $s_n$ is odd. You are given $F(10)=199$.

Find $F({10}^{16})$ giving your answer modulo ${10}^9+7$.