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Problem 760

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Sum over Bitwise Operators

Define
$${\displaystyle g(m,n)=(m\oplus n)+(m\vee n)+(m\wedge n)}$$
where $\oplus ,\vee ,\wedge$ are the bitwise XOR, OR and AND operator respectively.

Also set
$${\displaystyle G(N)=\sum _{n=0}^N\sum _{k=0}^{n}g(k,n-k)}$$

For example, $G(10)=754$ and $G({10}^2)=583766$.

Find $G({10}^{18})$. Give your answer modulo $1000000007$.

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位运算求和

记
$${\displaystyle g(m,n)=(m\oplus n)+(m\vee n)+(m\wedge n)}$$
其中$\oplus ,\vee ,\wedge$分别代表“按位异或”、“按位或”和“按位与”运算。

再记
$${\displaystyle G(N)=\sum _{n=0}^N\sum _{k=0}^{n}g(k,n-k)}$$

例如，$G(10)=754$，$G({10}^2)=583766$。

求$G({10}^{18})$，并将你的答案对$1000000007$取余。

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