## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 307-318 |

Number of pages | 12 |

Journal | Australasian Journal of Combinatorics |

Volume | 30 |

State | Published - Dec 1 2004 |

Externally published | Yes |