Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers

TL;DR

This paper derives closed form equations to directly compute triangular numbers that are multiples of other triangular numbers, covering different cases based on the multiplier's properties, thus avoiding recursive calculations.

Contribution

It introduces explicit formulas for calculating such triangular numbers for multipliers of ranks 1 to 4, expanding beyond previous recursive methods.

Findings

Closed form equations for four cases of multipliers.

Examples for non-square multipliers 2, 3, 5, and 8.

Direct computation without recursion.

Abstract

Triangular numbers that are multiple of other triangular numbers are investigated. It is known that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers which are triangular numbers. If the multiplier is a squared integer, there is either one or no solution, depending on the multiplier value. Instead of recurrent relations, we develop in this paper closed form equations to calculate directly the values of triangular numbers and their indices without the need of knowing the previous solutions. We develop the theoretical equations for four cases of ranks from 1 to 4 and we give several examples for non-square multipliers 2, 3, 5 and 8.