Minimal free resolution of generalized repunit algebras

Let $\mathbb{k}$ be an arbitrary field and let $b>1,n>1$ and $a$ be three positive integers. In this paper we explicitly describe a minimal $S-$graded free resolution of the semigroup algebra $\mathbb{k}[S]$ when $S$ is a generalized repunit numerical semigroup, that is, when $S$ is the submonoid of $\mathbb{N}$ generated by $\{a_1,a_2,\dots ,a_n\}$ where $a_1={\sum }_{j=0}^{n-1}{b}^j$ and $a_i-a_{i-1}=a{b}^{i-2},i=2,\dots ,n$, with $\gcd (a,a_1)=1$.