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 p3∆(n), the number of partitions of the integer n into three triangular numbers, as well as p3∆(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, p3∆(n) and p3∆(n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p3∆(27n+ 12) = 3p3∆(3n+ 1) and p3∆(27n+ 12) = 3p d 3∆(3n+ 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.