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A063984 Minimal number of integer points in the Euclidean plane which are contained in any convex n-gon whose vertices have integer coordinates. 2
0, 0, 1, 1, 4, 4, 7, 10, 17, 19, 27, 34, 45, 52 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,5

COMMENTS

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.

Sequence continues 0, 0, 1, 1, 4, 4, 7, 10, 17, 19, 27, 34, 45, 52, [66-72], 79, [96-105], 112, [133-154], 154

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.

LINKS

Table of n, a(n) for n=3..16.

I. Barany and N. Tokushige, The minimum area of convex lattice n-gons, Combinatorica, 24 (No. 2, 2004), 171-185.

Tian-Xin Cai, On the minimum area of convex lattice polygons, Taiwanese Journal of Mathematics, Vol 1, No 4 (1997).

W. Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518.

C. J. Colbourn, R. J. Simpson, A note on bounds on the minimum area of convex lattice polygons, Bull. Austral. Math. Soc. 45 (1992) 237-240.

S. Rabinowitz, O(n^3) bounds for the area of a convex lattice n-gon, Geombinatorics, vol. II, 4(1993), p. 85-88.

R. J. Simpson, Convex lattice polygons of minimum area, Bulletin of the Australian Math. Society, 42 (1990), p. 353-367.

FORMULA

A070911(n)/2 = a(n) + n/2 - 1. [Simpson]

See Barany & Norihide for asymptotics.

EXAMPLE

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.

CROSSREFS

Cf. A070911.

Sequence in context: A109544 A187893 A293678 * A211643 A284640 A036605

Adjacent sequences:  A063981 A063982 A063983 * A063985 A063986 A063987

KEYWORD

nice,more,nonn

AUTHOR

Pierre Bornsztein (pbornszt(AT)club-internet.fr), Sep 06 2001; May 20 2002

EXTENSIONS

Additional comments from Steven Finch, Dec 06 2003

More terms from Matthias Henze, Jul 27 2015

STATUS

approved

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Last modified October 23 18:36 EDT 2018. Contains 316529 sequences. (Running on oeis4.)