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 $10^{16}$.
Solution
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