Representing Sets with Sums of Triangular Numbers

We investigate here sums of triangular numbers f(x) := Sigma(i)b(i)T(xi) where T-n is the nth triangular number. We show that for a set of positive integers S, there is a finite subset S-0 such that f represents S if and only if f represents S-0. However, computationally determining S-0 is ineffective for many choices of S. We give an explicit and efficient algorithm to determine the set S-0 under certain generalized Riemann hypotheses, and implement the algorithm to determine S-0 when S is the set of all odd integers.