The Frobenius problem for numerical semigroups generated by sequences that satisfy a linear recurrence relation

Abstract

Consider a sequence of positive integers of the form $c{a}^n-d$, $n\ge 1$, where $a,c$ and $d$ are positive integers, $a>1$. For each $n\ge 1$, let $S_n$ be the submonoid of $\mathbb{N}$ generated by ${\mathbf{s}}_j=c{a}^{n+j}-d$, with $j\in \mathbb{N}$. We obtain a numerical semigroup $(1/e)S_n$ by dividing every element of $S_n$ by $e=\gcd (S_n)$. We characterize the embedding dimension of $S_n$ and describe a method to find the minimal generating set of $S_n$. We also show how to find the maximum element of the Apéry set $\mathrm{Ap}(S_n,{\mathbf{s}}_0)$, characterize the elements of $\mathrm{Ap}(S_n,{\mathbf{s}}_0)$, and use these results to compute the Frobenius number of the numerical semigroup $(1/e)S_n$, where $e=\gcd (S_n)$.

arXiv comment30 pages