A combinatorial approach to Frobenius numbers of some special sequences

Abstract

Let $A=(a_1,a_2,\dots ,a_n)$ be relative prime positive integers with $a_i\ge 2$. The Frobenius number $g\left(A\right)$ is the greatest integer not belonging to the set $\left\{{\sum }_{i=1}^n{a}_i{x}_i\right|x_i\in N\}$. The general Frobenius problem includes the determination of $g\left(A\right)$ and the related Sylvester number $n\left(A\right)$ and Sylvester sum $s\left(A\right)$. We present a combinatorial approach to the Frobenius problem. Basically, we transform the problem into an easier optimization problem. If the new problem can be solved explicitly, then we obtain a formula for $g\left(A\right)$. We illustrate the idea by giving concise proofs and extensions of several existing formulas, as well as new formulas for $g\left(A\right),n\left(A\right),s\left(A\right)$. Moreover, we give a generating function approach to $n\left(A\right),s\left(A\right)$, and even to the more general Sylvester power sum.