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A063984
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Minimal number of integer points in the Euclidean plane which are contained in the interior of any convex n-gon whose vertices have integer coordinates.
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8
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0, 0, 1, 1, 4, 4, 7, 10, 17, 19, 27, 34, 45, 52, 68, 79, 98, 112, 135, 154, 183, 199, 237, 262, 300, 332, 378, 416, 469, 508, 573, 616, 688, 732, 818, 872, 959, 1020, 1120, 1202, 1305, 1391, 1504, 1598, 1724, 1815, 1961, 2064, 2220, 2332, 2497, 2625, 2785
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OFFSET
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3,5
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COMMENTS
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Consider convex lattice n-gons, that is, polygons whose n vertices are points on the integer lattice Z^2 and whose interior angles are strictly less than Pi. a(n) is the least possible number of lattice points in the interior of such an n-gon.
The result a(5) = 1 seems to be due to Ehrhart. Using Pick's formula, it is not difficult to prove that the determination of a(k) is equivalent to the determination of the minimal area of a convex k-gon whose vertices are lattice points.
Results before 2018 for odd n came from the following authors: a(3) (trivial), a(5) (Arkinstall), a(7) and a(9) (Rabinowitz), a(11) (Olszewska), a(13) (Simpson) and a(15) (Castryck). - Jamie Simpson, Oct 18 2022
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LINKS
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FORMULA
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a(n) = A070911(n)/2 - n/2 + 1. [Simpson]
See Barany & Tokushige for asymptotics.
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EXAMPLE
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For example, every convex pentagon whose vertices are lattice points contains at least one lattice point and it is not difficult to construct such a pentagon with only one interior lattice point. Thus a(5) = 1.
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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Pierre Bornsztein (pbornszt(AT)club-internet.fr), Sep 06 2001; May 20 2002
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EXTENSIONS
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a(26)-a(55) from and definition clarified by Günter Rote, Sep 19 2023
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STATUS
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approved
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