The Frobenius problem for special progressions

<abstract><p>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+db^2, \ldots, ha+db^k\} $. In this paper, for $ S_c: = \{a, ha+d, ha+c+db, ha+2c+db^2, \ldots, ha+kc+db^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.</p></abstract>