Triangular and square triangular numbers involving generalized pell numbers

Abstract

Triangular numbers denoted by T-n are the numbers of the form T-n = n(2+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 S-n. One can write Sn as S-n = s(n)(2) =t(n)(t(n)+1)/2 where s(n) and to 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 T-n. = S-n, holds by using s(n), and t(n). We also deduce some formulas on perfect squares, sums of s(n), t(n), S-n,T-n, divisibility properties and integer solutions of Pell equations.