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Problem 725

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Digit sum numbers

A number where one digit is the sum of the other digits is called a digit sum number or DS-number for short. For example, $352$, $3003$ and $32812$ are DS-numbers.

We define $S(n)$ to be the sum of all DS-numbers of $n$ digits or less.

You are given $S(3)=63270$ and $S(7)=85499991450$.

Find $S(2020)$. Give your answer modulo ${10}^{16}$.

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数字和数

若一个数中有一位数字是其它数字之和，则称这个数为数字和数或简称为DS-数。例如，$352$、$3003$和$32812$都是DS-数。

记$S(n)$为所有$n$位及$n$位以下的DS-数之和。

已知$S(3)=63270$，$S(7)=85499991450$。

求$S(2020)$，并将你的答案对${10}^{16}$取余。

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