We resolve the open problem of characterizing the Frobenius number g(A) for shifted square sequences \(A = (a, a+1^2, \ldots , a+k^2)\) with positive integer a, confirming a conjecture of Einstein et al. (Integers 7:A15, 2007). By combining a combinatorial reduction to an optimization problem with Lagrange’s Four-square theorem and generating function techniques, we derive a semi-explicit formula for g(A): a piecewise quadratic polynomial in a, classified by residue classes modulo \(k^2\) .
