A product rule for triangular numbers
by David Radcliffe
The nth triangular number is . 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)
The triangular numbers satisfy many interesting properties, including a product rule:
This rule can be demonstrated visually by subdividing a triangle into smaller triangles. The following picture illustrates the case
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.