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Problem 912

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Where are the Odds?

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 ${10}^9+7$.

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奇数在哪里？

令$s_n$为第$n$个二进制表示不包含三个连续的$1$的正整数。

例如，$s_1=1$，$s_7=8$。

定义$F(N)$为所有满足$n\le N$且$s_n$为奇数的$n^2$之和。已知$F(10)=199$。

求$F({10}^{16})$，并对${10}^9+7$取余作为你的答案。

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