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 $1\,000\,000\,007$.

To solve this problem, we can calculate the values of `g(m, n)` and `G(N)` using the given formulas. Since `G(N)` depends on the values of `g(m, n)` for different pairs of `m` and `n`, we will start by defining a function to calculate `g(m, n)`.

“`python
MOD = 1000000007

def g(m, n):
return (m ^ n) + (m | n) + (m & n)
“`

Note that we have defined a constant `MOD` to take the modulo operation at every step to avoid integer overflow.

Next, we can define a function to calculate `G(N)` using a nested loop for the summation.

“`python
def G(N):
result = 0
for n in range(N + 1):
for k in range(n + 1):
result += g(k, n – k)
result %= MOD
return result
“`

Now, we can simply call the `G(N)` function with `N = 10**18` and print the result.

“`python
N = 10**18
result = G(N)
print(result)
“`

Remember to run the code with a Python interpreter capable of handling such large numbers efficiently, or consider using a library like `numpy` or `numba` for faster computations.

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