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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.
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%I #15 Jun 23 2022 20:37:38

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

%N 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.

%C 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.

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

%H Michael Collier, Dmitry Kamenetsky, Herbert Kociemba, Tom Sirgedas, <a href="http://dxdy.ru/topic113868.html">Al Zimmermann - Polygonal Areas</a>. Discussion in newsgroup dxdy.ru.

%H Hugo Pfoertner, <a href="/A288247/a288247.pdf">Minimum Area Lattice Polygons</a>, Illustrations for n = 3 ... 11.

%H Markus Sigg, <a href="http://antiton.de/PolygonalAreas/index.html?(1,1),(6,5),(9,7),(7,6),(3,3),(4,4),(8,9),(2,2),(5,8)">JavaScript visualization of lattice polygons</a>. Example showing the minimal 9-sided polygon.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pick%27s_theorem">Pick's theorem</a>.

%H Al Zimmermann's Programming Contests, <a href="http://azspcs.com/Contest/PolygonalAreas">Polygonal Areas</a>.

%Y Cf. A288248, A288249.

%K nonn,more

%O 3,1

%A _Hugo Pfoertner_, Jun 07 2017