We investigate here sums of triangular numbers f(x) := P i biTxi where Tn is the n-th triangular number. We show that for a set of positive integers S there is a finite subset S0 such that f represents S if and only if f represents S0. However, computationally determining S0 is ineffective for many choices of S. We give an explicit and efficient algorithm to determine the setS0 under certain Generalized Riemann Hypotheses, and implement the algorithm to determine S0 when S is the set of all odd… Expand