Numerical Semigroups with unique Ap\'{e}ry expansions

In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by the partial sums of a geometric progression, in embedding dimension $4$. Both these classes have the common feature that they have unique expansions of the Ap'{e}ry set elements.