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 $G (17526 \times 10^{9})$ .

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 $G (17526 \times 10^{9})$ 。