Full Grid Lattice Polygons with Maximal Sum of Squares of Edge-Lengths

Abstract

Consider a subset [1,2,...,n]x[1,2,...,n] of the plane integer lattice. Take any non self-intersecting n^2-gon built on it (straight angles are allowed). The square of a side length is a positive integer. It is thus natural to ask how large the sum of square lengths of such an n^2-gon can be. This maximal value is a new integer sequence, labeled by A358212 in OEIS. In this note we give the lower bound and conjecture that this in fact is the correct answer. We further investigate proper n^2-gons (straight angles are not allowed) and present analogous results. Both sequences (conjecturally) have a different growth size.