A cubic triangular number is a positive integer that is simultaneously cubic and triangular. Such a number must therefore satisfy T_n = m^3 for some positive integers n and m, where T_n is a triangular number, so 1/2 n(n + 1) = m^3. But then (2n + 1)^2 - 1 = (2m)^3 or (2n + 1)^2 - (2m)^3 = 1.

cubic number | square triangular number | triangular number