Expressing an integer as the sum of triangular numbers

Show HTML problem contentShow HTML problem content  Published on Sunday, 25th February 2018, 04:00 am; Solved by 477;
Difficulty rating: 35%

Problem 621

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 $G(17 526 \times 10^9)$.