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

Abstract

Let $a,b$ be positive integers. In this note, we study the numerical semigroup $H=⟨a,a+1,b⟩$ 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 $R$.

14 pages, Comments welcome