On Frobenius Numbers of Shifted Power Sequences

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