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Inverse Digit Sum

Published on Saturday, 19th October 2019, 04:00 pm; Solved by 408

Problem 684

Define $s(n)$ to be the smallest number that has a digit sum of $n$. For example $s(10) = 19$.
Let $\displaystyle S(k) = \sum_{n=1}^k s(n)$. You are given $S(20) = 1074$.

Further let $f_i$ be the Fibonacci sequence defined by $f_0=0, f_1=1$ and $f_i=f_{i-2}+f_{i-1}$ for all $i \ge 2$.

Find $\displaystyle \sum_{i=2}^{90} S(f_i)$. Give your answer modulo $1\,000\,000\,007$.