Square triangular number

In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: Write Nk for the kth square triangular number, and write sk and tk for the sides of the corresponding square and triangle, so that Define the triangular root of a triangular number N = n(n + 1)/2 to be n. From this definition and the quadratic formula, Therefore, N is triangular (n is an integer) if and only if 8N + 1 is square. Consequently, a square number M2 is also triangular if and only if 8M2 + 1 is square, that is, there are numbers x and y such that x2 − 8y2 = 1. This is an instance of the Pell equation with n = 8. All Pell equations have the trivial solution x = 1, y = 0 for any n; this is called the zeroth solution, and indexed as (x0, y0) = (1,0). If (xk, yk) denotes the kth nontrivial solution to any Pell equation for a particular n, it can be shown by the method of descent that Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever n is not a square. The first non-trivial solution when n = 8 is easy to find: it is (3,1). A solution (xk, yk) to the Pell equation for n = 8 yields a square triangular number and its square and triangular roots as follows: Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from 6 × (3,1) − (1,0) = (17,6), is 36. The sequences Nk, sk and tk are the OEIS sequences OEIS: A001110, OEIS: A001109, and OEIS: A001108 respectively. In 1778 Leonhard Euler determined the explicit formula:12–13