Triangular numerical semigroups (Q6067751)

Let \(\mathbb{N}\) denote the set of non-negative integers. A numerical semigroup is an additive submonoid of \(\mathbb{N}\) with finite complement in \(\mathbb{N}\). Every numerical semigroup \(S\) is minimally generated by \(S^*\setminus(S^*+S^*)\). The cardinality of this set is called the embedding dimension of \(S\). For \(n\in\mathbb{N}\), let \(t_n=\binom{n+1}2\) and let \(S_n\) be the submonoid of \(\mathbb{N}\) generated by \(\{t_{n+k}: k\in\mathbb{N} \}\). Then, \(S_n\) is a numerical semigroup. By analizying recurrence relations for the sequence \(\{t_n\}_{n\in\mathbb{N}}\), the authors give an explicit description of \(S_n^*\setminus(S_n^*+S_n^*)\), which is a subset of \(\{t_k: k\in\{n,n+1,\dots,2n\}\}\). This enables them to provide upper and lower bounds for the embedding dimension of \(S_n\).