Triangular and square triangular numbers involving generalized pell numbers

Abstract

Triangular numbers denoted by Tn are the numbers of the form Tn = n(n+1)/2 for n ≥ 0. There are infinitely many triangular numbers that are also square numbers. These numbers are called square triangular numbers and denoted by Sn. One can write Sn as where sn and tn denote the sides of the corresponding square and triangle. In this work, we derive some algebraic identities on triangular, square triangular numbers and also squares and triangles. We construct a connection between triangular and square triangular numbers. We determine when the equality Tn = Sn holds by using sn and tn. We also deduce some formulas on perfect squares, sums of sn,tn,Sn,Tn, divisibility properties and integer solutions of Pell equations.