The Frobenius problem for special progressions

Let $S$ be a given finite set of positive and relatively prime integers. Denote $L(S)$ to be the set of integers obtained by taking all nonnegative integer linear combinations of integers in $S$. It is well known that there are finitely many positive integers that are not in $L(S)$. Let $g(S)$ and $n(S)$ represent the greatest integer that does not belong to $L(S)$ and the number of nonnegative integers that do not belong to $L(S)$, respectively. The Frobenius problem is to determine $g(S)$ and $n(S)$. In 2016, Tripathi obtained results on $g(S)$ and $n(S)$ when $S=\{a,ha+d,ha+db,ha+d{b}^2,\dots ,ha+d{b}^k\}$. In this paper, for $S_c:=\{a,ha+d,ha+c+db,ha+2c+d{b}^2,\dots ,ha+kc+d{b}^k\}$ with $h,c$ being nonnegative integers, $a,b,d$ being positive integers and $gcd(a,d)=1$, we focused the investigation on formulas for $g(S_c)$ and $n(S_c)$. Actually, we gave formulas for $g(S_c)$ and $n(S_c)$ for all sufficiently large values of $d$ when $c$ is any multiple of $d$ or certain multiples of $a$. This generalized the results of Tripathi in 2016.