Partitions into three triangular numbers

A celebrated result of Gauss states that every positive integer can be represented as the sum of three triangular numbers. In this article we study p_3Δ(n), the number of partitions of the integer n into three triangular numbers, as well as p^d_3Δ(n), the number of partitions of n into three distinct triangular numbers. Unlike t(n), which counts the number of representations of n into three triangular numbers, p_3Δ(n) and p^d_3Δ(n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p_3Δ(27n + 12) = 3p_3Δ(3n + 1) and p^d_3Δ(27n + 12) = 3p^d_3Δ(3n + 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.