triangular square number

A number which is simultaneously square and triangular. Let T_n denote the nth triangular number and S_m the mth square number, then a number which is both triangular and square satisfies the equation T_n = S_m, or 1/2 n(n + 1) = m^2. Completing the square gives 1/2(n^2 + n) | = | 1/2 (n + 1/2)^2 - (1/2)(1/4) | = | m^2 1/8 (2n + 1)^2 - 1/8 | = | m^2 (2n + 1)^2 - 8m^2 | = | 1.

cubic triangular number | pentagonal square number | pentagonal square triangular number | square number | square root | triangular number