Let \(A_k= \{a_1, \dots, a_k\}\) be a set of natural numbers with \(\text{gcd} (a_1, \dots, a_k)=1\). The greatest integer \(g= g(A_k)\) with no representation \(g= \sum^k_{i=1} x_i a_i\), \(x_i\in \mathbb{N}_0= \{0, 1, \dots\}\) is called the Frobenius number of \(A_k\). For a special class of sets \(A_k\) we transform the problem of determining \(g(A_k)\) to an equivalent one and get an algorithm whose number of arithmetical operations for computing \(g(A_k)\) depends on \(k\) only. Furthermore some new formulas for \(g(A_k)\) in special cases are given.