Families of numerical semigroups and a special case of the Huneke-Wiegand conjecture

The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. García-Sánchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings $\mathbb{k}[\mathrm{\Gamma }]$ can be answered in the affirmative by locating certain arithmetic sequences within the numerical semigroup $\mathrm{\Gamma }$. In this paper, we use their approach to prove the Huneke-Wiegand conjecture in the case where $\mathrm{\Gamma }$ is generated by a generalized arithmetic sequence and showcase how visualizations can be leveraged to find the requisite arithmetic sequences.