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 p3Δ(n), the number of partitions of the integer n into three triangular numbers, as well as pd3Δ(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, p3Δ(n) and pd3Δ(n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p3Δ(27n + 12) = 3p3Δ(3n + 1) and pd3Δ(27n + 12) = 3pd3Δ(3n + 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.
|Original language||English (US)|
|Number of pages||12|
|Journal||Australasian Journal of Combinatorics|
|State||Published - Dec 1 2004|