A positive integer with $k$ (decimal) digits is called balanced if its first $\lceil k/2 \rceil$ digits sum to the same value as its last $\lceil k/2 \rceil$ digits, where $\lceil x \rceil$, pronounced ceiling of $x$, is the smallest integer $\ge x$, thus $\lceil \pi \rceil = 4$ and $\lceil 5 \rceil = 5$.
So, for example, all palindromes are balanced, as is $13722$.
Let $T(n)$ be the sum of all balanced numbers less than $10^n$.
Thus: $T(1) = 45$, $T(2) = 540$ and $T(5) = 334795890$.
Find $T(47) \bmod 3^{15}$.
Solution
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