Frobenius numbers and the first Hilbert coefficients of certain numerical semigroup rings

Let $a,b$ be positive integers. In this note, we study the numerical semigroup $H=\left<a,a+1,b\right>$ and and the associated numerical semigroup ring $R=k[[H]]$. Under the certain conditions, we provide explicit formulas for the Frobenius number of $H$ and for the first Hilbert coefficient of $R$.

💡 Research Summary

The paper investigates the numerical semigroup (H=\langle a, a+1, a+d\rangle) (with (1<d<a)) and its associated semigroup ring (R=k