Partitions into three triangular numbers

Michael D. Hirschhorn, James A. Sellers

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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

Original languageEnglish (US)
Pages (from-to)307-318
Number of pages12
JournalAustralasian Journal of Combinatorics
StatePublished - 2004
Externally publishedYes


Dive into the research topics of 'Partitions into three triangular numbers'. Together they form a unique fingerprint.

Cite this