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Problem 217

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Balanced Numbers

A positive integer with k (decimal) digits is called balanced if its first ⌈^k/₂⌉ digits sum to the same value as its last ⌈^k/₂⌉ digits, where ⌈x⌉, pronounced ceiling of x, is the smallest integer ≥ x, thus ⌈π⌉ = 4 and ⌈5⌉ = 5.

So, for example, all palindromes are balanced, as is 13722.

Let T(n) be the sum of all balanced numbers less than 10ⁿ.
Thus: T(1) = 45, T(2) = 540 and T(5) = 334795890.

Find T(47) mod 3¹⁵.

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平衡数

如果一个k位（十进制）正整数，其前⌈^k/₂⌉个数字之和等于其后⌈^k/₂⌉个数字之和，则称之为平衡数。这里⌈x⌉表示x的上取整函数，也就是≥ x的最小整数，例如⌈π⌉ = 4以及⌈5⌉ = 5。

举例来说，所有的回文数都是平衡数；13722也是平衡数。

记T(n)是所有小于10ⁿ的平衡数之和。
因此：T(1) = 45，T(2) = 540以及T(5) = 334795890。

求T(47) mod 3¹⁵。

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