William Thurston asks if there is a characterization of numbers that can be expressed as the sum of two triangles (A020756), in the spirit of the characterization of a sum of two squares (A001481). Emeric Deutsch earlier submitted that the sequence in question was A051533, the sums of two positive triangular numbers. This is a much more difficult beastie to characterize, and I don't think it was intended. Anyway, Fred Helenius correctly characterized A020756 as the numbers n such that 4n+1 is in A001481. Thus characterizations of A001481 transform to characterizations of A001481. Specifically %C A001481 Closed over multiplication. [Not in OEIS] %C A020756 4n+1 = odd square * product of distinct primes of form 4k+1. %C A020756 Closed over f(x, y) = 4xy + x + y. Another nice property noted on A020756 is that the sums of two triangles are precisely the sums of a square and a pronic.