TRIANGULAR SQUARE AND TRIANGULAR PRONIC NUMBERS
Numbers which are both triangular and square are called triangular squares, numbers which are both triangular and pronic are called triangular pronics.
TRIANGULAR, SQUARE, AND PRONIC
Inputting the next counting number into the formulae above outputs the next number in either the triangular, square, or pronic sequence. The respective m-valued or n-valued number correlates to the sequential order of the term in the series.
ALTERNATING SEQUENCES
Above is a diagram which maps out the first 4 triangular square and triangular pronic numbers as well as their respective m and n-values. These two series when put in ascending order, alternate back and forth between triangular square and triangular pronic values.
THE FIRST 49 TRIANGULAR NUMBERS
Above are the first 49 triangular numbers inscribed within 35 concentric even-sided polygons. While pronic and square numbers line up in a row, triangular numbers form a galaxy-like spiral.