The Frobenius formula for \(A=(a,ha+d,ha+b_2d, \ldots ,ha+b_kd)\)

Given a set of positive integers \(A=(a_1, a_2, \ldots , a_n)\) $A=(a_1,a_2,\dots ,a_n)$ whose greatest common divisor is 1, the Frobenius number g(A) is the largest integer not representable as a linear combination of the \(a_i\) $a_i$ ’s with nonnegative integer coefficients. We find the stable property introduced for the square sequence \(A=(a,a+1,a+2^2,\dots , a+k^2)\) $A=(a,a+1,a+2^2,\cdots ,a+k^2)$ naturally extends for \(A(a)=(a,ha+dB)=(a,ha+d,ha+b_2d, \ldots ,ha+b_kd)\) $A(a)=(a,ha+dB)=(a,ha+d,ha+b_{2}d,\dots ,ha+b_{k}d)$ . This gives a parallel characterization of g(A(a)) as a "congruence class function" modulo \(b_k\) $b_k$ when a is large enough. For orderly sequence \(B=(1,b_2,\dots ,b_k)\) $B=(1,b_2,\cdots ,b_k)$ , we find good bound for a. In particular we calculate \(g(a,ha+dB)\) $g(a,ha+dB)$ for \(B=(1,2,b,b+1)\) $B=(1,2,b,b+1)$ , \(B=(1,2,b,b+1,2b)\) $B=(1,2,b,b+1,2b)$ , \(B=(1,b,2b-1)\) $B=(1,b,2b-1)$ , and \(B=(1,2, \ldots ,k,K)\) $B=(1,2,\dots ,k,K)$ .