  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.
