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 $G(6 \times 5^N)$.