On the numerical semigroup generated by $\{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\}$

Let $b, n$ be two positive integers such that $b\geq 2$, and $S(b,n)$ be the numerical semigroup generated by $\{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\}$. Applying two order relations, we give formulas for computing the embedding dimension, the Frobenius number, the type and the genus of $S(b,n)$.