  We investigate here sums of triangular numbers $f(x):=\sum_i b_i T_{x_i}$
where $T_n$ is the $n$-th 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.

Number Theory