A product rule for triangular numbers

by David Radcliffe

The nth triangular number is T(n) = 1 + 2 + \ldots + n = \frac12 n(n+1). It represents the number of dots in a triangular arrangement, with 1 dot in the first row, 2 dots in the second row, etc. (Image source: Wikipedia)

Triangular numbers

The triangular numbers satisfy many interesting properties, including a product rule:

T(mn) = T(m)T(n) + T(m-1)T(n-1)

This rule can be demonstrated visually by subdividing a triangle into smaller triangles. The following picture illustrates the case

T(20) = T(5)T(4) + T(4)T(3).

Triangular Numbers: T(20) = T(5) T(4) + T(4) T(3)

Inspired by a question of James Tanton, I sought to find all sequences that satisfy this product rule. This problem has a lovely solution, and I encourage you to discover it for yourself. I will outline my solution in subsequent posts.