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Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers

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.