A product rule for triangular numbers

TL;DR

This paper characterizes all sequences satisfying a specific product rule related to triangular numbers, identifying exactly five such sequences including the triangular numbers themselves.

Contribution

It proves that only five sequences, including the triangular numbers, satisfy the given product rule for all positive integers.

Findings

Triangular numbers satisfy the product rule.

Exactly five sequences meet the product rule criteria.

The result characterizes the uniqueness of these sequences.

Abstract

We prove that there are exactly five sequences, including the triangular numbers, that satisfy the product rule $T(mn) = T(m) T(n) + T(m-1) T(n-1)$ for all $m, n \ge 1$.