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的和（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示为三个三角形数的和。

记$G(n)$为将$n$表示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$G(17526\times10^9)$。