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A070911
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a(n) is twice the least possible area enclosed by a convex lattice n-gon.
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9
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1, 2, 5, 6, 13, 14, 21, 28, 43, 48, 65, 80, 103, 118, 151, 174, 213, 242, 289, 328, 387, 420, 497, 548, 625, 690, 783, 860, 967, 1046, 1177, 1264, 1409, 1498, 1671, 1780, 1955, 2078, 2279, 2444, 2651, 2824, 3051, 3240, 3493, 3676, 3969, 4176, 4489, 4714, 5045, 5302, 5623, 5906
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OFFSET
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3,2
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COMMENTS
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A convex lattice n-gon is a polygon whose n vertices are points on the integer lattice Z^2 and whose interior angles are strictly less than Pi.
For the even-indexed values, see A089187. The precisely known odd values were a(3)=1 (trivial), a(5)=5 and a(7)=13 (Arkinstall), a(9)=21 (Rabinowitz), a(11)=43 (Olszewska), a(13)=65 (Simpson), and a(15)=103 (Castryck). Additional values up to a(25) were first obtained as upper bounds "by massive calculations with several independent search programs" by Hugo Pfoertner. Pfoertner has made nice pictures of the best polygons he has found. See his link below. - Jamie Simpson, Dec 08 2022, adapted by Günter Rote, Sep 18 2023
The values a(n) can be computed in time polynomial in n by an algorithm of Eppstein, Overmars, Rote, and Woeginger from 1992: They showed how to compute the smallest convex n-gon in a set of N points in O(nN^3) time. The N points can be taken as an O(n^2) X O(n^2) grid: Each dimension of the bounding rectangle must be at least n/2, because a horizontal or vertical line can contain at most two vertices; since the area is known to be bounded by n^3, the other dimension cannot exceed 4n^2. In our case, the runtime can be reduced to O(nN^2), since the lowest vertex can be assumed to be a fixed point, say, the origin. By considering the lattice width, the grid can be reduced to size N=O(n^2) X O(n^1.5). Overall, this yields a theoretical runtime bound of O(n^8), for reporting all k-gons up to size n. This estimate agrees roughly with the observed runtimes in practice.
I have implemented the algorithm in Python and uploaded the program to the Code Golf Stackexchange site. It runs up to a(40) in a couple of minutes and produces some smallest polygon for each n. The values up to a(102) have been computed on a workstation in 31 hours. (End)
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LINKS
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FORMULA
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a(n)/2 = A063984(n) + n/2 - 1. [Simpson]
See Bárány and Tokushige for asymptotics.
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PROG
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See Code Golf StackExchange link, and the link to the Python program.
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CROSSREFS
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See A089187 for the even-indexed subsequence. See A063984 for further information.
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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), May 20 2002
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EXTENSIONS
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a(13), a(26) and virtually all terms with even n up to a(42) (as given in A089187) go back to Jamie Simpson, Dec 07 2003
a(17)-a(26) restored and a(27) onwards added by Günter Rote, Sep 18 2023
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STATUS
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approved
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