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Problem #411

Standard Nim
 Public ★(x7) 3w:5d by Min_25 8xp Math 53.8%

Let $C(n, k)$ be the number of integer solutions $(x_1, \ldots, x_k)$ such that $\bigoplus_{i=1}^k x_i = 0$ and $0 \le x_i < n$ for each $i$. Here, $x \oplus y$ means $x$ xor $y$.

Let $F(n)$ be the $n$-th fibonacci number: $F(0) = 0, F(1) = 1$, and $F(i) = F(i - 1) + F(i - 2)$ for $i \ge 2$.

Let $$S(n) := \sum_{i=1}^{n} \sum_{j=1}^{n} C(F(i), F(j)).$$ You are given $S(6) = 2153296$ and $S(7) = 18998620089329$.

Find $S(92)$ modulo $10^9 + 7$.

[My timing] 0.3 seconds.

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