A288247 2 * smallest possible area of a simple n-sided lattice polygon whose vertex coordinates x and y are both independent permutations of the integers 1 ... n, subject to the condition that none of its edges are mutually parallel.

3, 2, 3, 8, 9, 7, 9, 8, 11, 12, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21

Offset

3,1

Comments

It is conjectured that a(n) = n-2 for all n > 15, i.e. the bound of Pick's theorem is achievable for all larger n.

Results are partially based on the discussion in the newsgroup dxdy.ru, see link.

Links

Table of n, a(n) for n=3..23.

Michael Collier, Dmitry Kamenetsky, Herbert Kociemba, Tom Sirgedas, Al Zimmermann - Polygonal Areas. Discussion in newsgroup dxdy.ru.

Hugo Pfoertner, Minimum Area Lattice Polygons, Illustrations for n = 3 ... 11.

Markus Sigg, JavaScript visualization of lattice polygons. Example showing the minimal 9-sided polygon.

Wikipedia, Pick's theorem.

Al Zimmermann's Programming Contests, Polygonal Areas.

Crossrefs

Cf. A288248, A288249.

Sequence in context: A131134 A151690 A346472 * A143744 A095243 A049921

Adjacent sequences: A288244 A288245 A288246 * A288248 A288249 A288250

Keyword

nonn,more

Author

Hugo Pfoertner, Jun 07 2017

Status

approved