A number which is simultaneously octagonal and square. Let O_n denote the nth octagonal number and S_m the mth square number, then a number which is both octagonal and square satisfies the equation O_n = S_m, or n(3n - 2) = m^2. Completing the square and rearranging gives (3n - 1)^2 - 3m^2 = 1.

octagonal number | square number