On Frobenius Numbers of Shifted Power Sequences

Abstract

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 $k^2$.

16 pages 2 tables

References in corpus (3)

• A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)
• The Frobenius Formula for $A=(a,ha+d,ha+b_{2}d,...,ha+b_{k}d)$
• The extended Frobenius problem for Fibonacci sequences incremented by a Fibonacci number