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Problem 684

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Inverse Digit Sum

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 $1000000007$.

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数字和的逆函数

记 $s(n)$为最小的数字和为$n$的数。例如$s(10)=19$。
记${\displaystyle S(k)=\sum _{n=1}^{k}s(n)}$。已知$S(20)=1074$。

记$f_i$为斐波那契数列，其中$f_0=0$，$f_1=1$，对任意$i\ge 2$有$f_i=f_{i-2}+f_{i-1}$。

求${\displaystyle \sum _{i=2}^{90}S(f_i)}$。将你的答案对$1000000007$取余。

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