For a positive integer n, consider the submonoid \(S_n\) of \({\mathbb {N}}\) generated by the sequence of positive integers \({\textbf{s}}_j=ca^{n+j}-d\), \(j\in {\mathbb {N}}\), where a, c, and d are integers, \(a\ge 2\), and c is positive. We unify ideas and results from previous works on specific cases for a, c, and d, and we prove some conjectures that arise in the general case. Under fairly general conditions on a, c and d, we characterize the embedding dimension of \(S_n\) in terms of a, c, and d, and provide a general characterization of the Apéry set \(\textrm{Ap}(S_n, {\textbf{s}}_0)\). We then derive formulas for the Frobenius number and genus of the numerical semigroup \((1/\lambda )S_n\), where \(\lambda =\gcd (S_n)\). We also estimate the set of pseudo-Frobenius numbers of \((1/\lambda )S_n\) and prove an inequality involving the type of \((1/\lambda )S_n\), which implies that Wilf’s conjecture is true for these numerical semigroups.

对于正整数n，考虑由正整数序列\({\textbf{ s}}_j=ca^{n+j 生成的 \({\mathbb { N}}\)的子幺半群\ (S_n\) }-d\)、\(j\in {\mathbb {N}}\)，其中a、  c和d为整数，\(a\ge 2\)且c为正数。我们统一了以前关于a、  c和d的具体情况的工作的想法和结果，并证明了一般情况下出现的一些猜想。在a、  c和d的相当一般的条件下，我们用a、  c和d来表征\(S_n\)的嵌入维数，并提供 Apéry 集\(\textrm{Ap}(S_n, {\textbf{s}}_0) 的一般表征\)。然后，我们推导数值半群\((1/\lambda )S_n\)的弗罗贝尼乌斯数和亏格公式，其中\(\lambda =\gcd (S_n)\)。我们还估计了\((1/\lambda )S_n\)的伪弗罗贝尼乌斯数集，并证明了涉及\((1/\lambda )S_n\)类型的不等式，这意味着 Wilf 的猜想对于这些数值半群。