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 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence in the OEIS).

Write N_k for the kth square triangular number, and write s_k and t_k for the sides of the corresponding square and triangle, so that

Define the triangular root of a triangular number ${\displaystyle N={\frac {n(n+1)}{2}}}$ to be ${\displaystyle n}$. From this definition and the quadratic formula, ${\displaystyle n={\frac {{\sqrt {8N+1}}-1}{2}}.}$ Therefore, ${\displaystyle N}$ is triangular if and only if ${\displaystyle 8N+1}$ is square. Consequently, a number ${\displaystyle M^{2}}$ is square and triangular if and only if ${\displaystyle 8M^{2}+1}$ is square, i. e., there are numbers ${\displaystyle x}$ and ${\displaystyle y}$ such that ${\displaystyle x^{2}-8y^{2}=1}$. This is an instance of the Pell equation, with ${\displaystyle n=8}$. All Pell equations have the trivial solution (1,0), for any n; this solution is called the zeroth, and indexed as ${\displaystyle (x_{0},y_{0})}$. If ${\displaystyle (x_{k},y_{k})}$ denotes the k'th non-trivial solution to any Pell equation for a particular n, it can be shown by the method of descent that ${\displaystyle x_{k+1}=2x_{k}x_{1}-x_{k-1}}$ and ${\displaystyle y_{k+1}=2y_{k}x_{1}-y_{k-1}}$. 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 ${\displaystyle (x_{k},y_{k})}$ to the Pell equation for n=8 yields a square triangular number and its square and triangular roots as follows: ${\displaystyle s_{k}=y_{k},t_{k}={\frac {x_{k}-1}{2}},}$ and ${\displaystyle N_{k}=y_{k}^{2}.}$ Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from (17,6) (=6×(3,1)-(1,0)), is 36.

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