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Problem 621


Problem 621


Expressing an integer as the sum of triangular numbers

Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including 0 as the lowest triangular number). In fact most numbers can be expressed as a sum of three triangular numbers in several ways.

Let $G(n)$ be the number of ways of expressing $n$ as the sum of three triangular numbers, regarding different arrangements of the terms of the sum as distinct.

For example, $G(9)=7$, as 9 can be expressed as: 3+3+3, 0+3+6, 0+6+3, 3+0+6, 3+6+0, 6+0+3, 6+3+0.
You are given $G(1000)=78$ and $G(10^6)=2106$.

Find $G(17526\times10^9)$.


将整数表示为三角形数的和

众所周知高斯证明了任意正整数可以被表示为三个三角形数(包括0这一最小的三角形数)。事实上大多数整数可以以多种方式被表达为三个三角形数的和。

记$G(n)$为将$n$表达为三个三角形数的方式数目,不同的排列视为不同的方式。

例如,$G(9)=7$,因为9可以被表示成:3+3+3,0+3+6,0+6+3,3+0+6,3+6+0,6+0+3,6+3+0。
已知$G(1000)=78$以及$G(10^6)=2106$。

求$G(17526\times10^9)$。