On Sylvester sums of compound sequence semigroup complements

In this paper, we consider the set $\mathit{\text{NR}}(G)$ of natural numbers which are not in the numerical semigroup generated by a compound sequence $G$. We generalize a result of Tuenter which completely characterizes $\mathit{\text{NR}}(G)$. We use this result to compute Sylvester sums, and we give a direct application to the computation of weights of higher-order Weierstrass points on some families of complex algebraic curves.