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

Author

Hugo Pfoertner, Jun 07 2017

Keywords

• nonn
• more

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

• 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.