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 ca n − d, n ≥ 1, where a, c and d are positive integers, a > 1. For each n ≥ 1, let S n be the submonoid of N generated by s j = ca n+j − d, with j ∈ 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 Ap(S n , s 0 ), characterize the elements of Ap(S n , s 0 ), and use these results to compute the Frobenius number of the numerical semigroup (1/e)S n , where e = gcd(S n ).

The Frobenius Number for Sequences of Triangular Numbers Associated with Number of Solutions

“…When b = 2 in Thabit numerical semigroups of the first kind or k = 1 in generalized Thabit, Thabit numerical semigroups are studied in [13]. Another similar type is on repunit numerical semigroups {(b n+i − 1)/(b − 1)|i ∈ N 0 } ( [14]) and more general types are considered in [2].…”

“…Since the considered sequence is infinite in[2,13,14,15,17,18,19], the result can be different even when p = 0.…”