{"app":{"locale":"en-GB"},"problem":{"content":"

The Fibonacci numbers $\\{f_n, n \\ge 0\\}$ are defined recursively as $f_n = f_{n-1} + f_{n-2}$ with base cases $f_0 = 0$ and $f_1 = 1$.

Define the polynomials $\\{F_n, n \\ge 0\\}$ as $F_n(x) = \\displaystyle{\\sum_{i=0}^n f_i x^i}$.

For example, $F_7(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + 13x^7$, and $F_7(11) = 268\\,357\\,683$.

Let $n = 10^{15}$. Find the sum $\\displaystyle{\\sum_{x=0}^{100} F_n(x)}$ and give your answer modulo $1\\,307\\,674\\,368\\,000 \\ (= 15!)$.

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