Consider the Fibonacci sequence $\{1,2,3,5,8,13,21,\ldots\}$.
We let $f(n)$ be the number of ways of representing an integer $n\ge 0$ as the sum of different Fibonacci numbers.
For example, $16 = 3+13 = 1+2+13 = 3+5+8 = 1+2+5+8$ and hence $f(16) = 4$.
By convention $f(0) = 1$.
Further we define $$S(n) = \sum_{k=0}^n f(k).$$ You are given $S(100) = 415$ and $S(10^4) = 312807$.
Find $\displaystyle S(10^{13})$.
Solution
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