 200px|thumb|square triangular number 36 depicted as a triangular number and as a square 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 an infinite number of square triangular numbers; the first few are 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 . ==explicit formulas== 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 :n_k = s_k^2 = \frac{t_k(t_k+1)}{2}. the sequences n'k, s'k and t'k are the oeis sequences , , and  respectively. in 1778 leonhard euler determined the explicit formula    :n_k = \left( \frac{(3 + 2\sqrt{2})^k - (3 - 2\sqrt{2})^k}{4\sqrt{2}} \right)^2. other equivalent formulas (obtained by expanding this formula) that may be convenient include : \begin{align} n_k &= {1 \over 32} \left( ( 1 + \sqrt{2} )^{2k} - ( 1 - \sqrt{2} )^{2k} \right)^2 = {1 \over 32} \left( ( 1 + \sqrt{2} )^{4k}-2 + ( 1 - \sqrt{2} )^{4k} \right) \\ &= {1 \over 32} \left( ( 17 + 12\sqrt{2} )^k -2 + ( 17 - 12\sqrt{2} )^k \right). \end{align} the corresponding explicit formulas for s'k and t'k are  : s_k = \frac{(3 + 2\sqrt{2})^k - (3 - 2\sqrt{2})^k}{4\sqrt{2}} and : t_k = \frac{(3 + 2\sqrt{2})^k + (3 - 2\sqrt{2})^k - 2}{4}. ==pell's equation== the problem of finding square triangular numbers reduces to pell's equation in the following way.  every triangular number is of the form t(t + 1)/2. th
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