

A063984


Minimal number of integer points in the Euclidean plane which are contained in any convex ngon whose vertices have integer coordinates.


2



0, 0, 1, 1, 4, 4, 7, 10, 17, 19, 27, 34, 45, 52
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OFFSET

3,5


COMMENTS

Consider convex lattice ngons, 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 ngon.
Sequence continues 0, 0, 1, 1, 4, 4, 7, 10, 17, 19, 27, 34, 45, 52, [6672], 79, [96105], 112, [133154], 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 kgon 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 ngons, Combinatorica, 24 (No. 2, 2004), 171185.
TianXin 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. 496518.
S. Rabinowitz, O(n^3) bounds for the area of a convex lattice ngon, Geombinatorics, vol. II, 4(1993), p. 8588.
R. J. Simpson, Convex lattice polygons of minimum area, Bulletin of the Australian Math. Society, 42 (1990), p. 353367.


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)clubinternet.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



