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

Given a set of positive integers A=(a_1, a_2, … , 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 ’s with nonnegative integer coefficients. We find the stable property introduced for the square sequence A=(a,a+1,a+2^2,… , a+k^2) naturally extends for A(a)=(a,ha+dB)=(a,ha+d,ha+b_2d, … ,ha+b_kd) . This gives a parallel characterization of g(A(a)) as a "congruence class function" modulo b_k when a is large enough. For orderly sequence B=(1,b_2,… ,b_k) , we find good bound for a. In particular we calculate g(a,ha+dB) for B=(1,2,b,b+1) , B=(1,2,b,b+1,2b) , B=(1,b,2b-1) , and B=(1,2, … ,k,K) .