Frobenius number of Almost Symmetric numerical generalized almost arithmetic semigroups

Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a,ha+d,ha+2d,…,ha+kd,c, that is its embedding dimension is k+2. In a previous work, the authors described the Apéry set and a Gröbner basis of the ideal defining S under one technical assumption, the complete version will be published in a forthcoming paper. In this paper we continue with this assumption and we describe the Pseudo Frobenius set. As a consequence we give a complete description of S when it is symmetric or almost symmetric as well as generalize and extend the previous results of Ignacio García-Marco, J. L. Ramírez Alfonsín and O. J. Rødseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. García-Sánchez. Moreover, for given numbers a, d, k, h, c, a simple algorithm allows us to determine if S is almost symmetric or not and furthermore to find its type and Frobenius number.