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

TL;DR

This paper investigates the Frobenius problem for a class of numerical semigroups generated by sequences satisfying linear recurrence relations, providing characterizations and methods to compute key invariants like the Frobenius number.

Contribution

It introduces a framework to analyze semigroups generated by sequences of the form ca^n - d, including methods to determine minimal generators and Frobenius numbers.

Findings

Characterization of the embedding dimension of S_n

Method to find minimal generating set of S_n

Procedure to compute Frobenius number of (1/e)S_n

Abstract

Consider a sequence of positive integers of the form $ca^n-d$, $n\geq 1$, where $a, c$ and $d$ are positive integers, $a>1$. For each $n\geq 1$, let $S_n$ be the submonoid of $\mathbb N$ generated by $\mathbf s_j=ca^{n+j}-d$, 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 Ap\'ery set ${\rm Ap}(S_n, \mathbf s_0)$, characterize the elements of ${\rm Ap}(S_n, \mathbf s_0)$, and use these results to compute the Frobenius number of the numerical semigroup $(1/e)S_n$, where $e=\gcd(S_n)$.