On Frobenius Formulas of Power Sequences

Abstract: Let $A=(a_1, a_2, \ldots, a_n)$ be a sequence of relative prime integers, each greater than or equal to 2. The Frobenius number $g(A)$ represents the largest integer that cannot be expressed as a nonnegative linear combination of the elements in $A$. Our recent work has introduced a combinatorial method for calculating $g(A)$ in the specific form $A=(a, a+B)=(a, a+b_1, \ldots, a+b_k)$. This method reduces the problem to a simpler optimization problem, denoted as $O_B(M)$. In this paper, we apply this method to solve the open problem of characterizing $g(A)$ for the square sequence $A=(a,a+1,a+2^2,\ldots, a+k^2)$. We demonstrate that the Frobenius number $g(a, a+1^2, a+2^2, \ldots)$ for the infinite square sequence admits a simpler solution. This is because the corresponding optimization problem, $O_B(M)$, can be solved by applying number-theoretic principles, specifically Lagrange's Four-Square Theorem and related results. Consequently, we are able to resolve the open problem by utilizing Lagrange's Four-Square Theorem in conjunction with generating functions.