3,5

We look at 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, [15-17], 19, 27, 34, [43-48], 52

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.

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.

Barany & Norihide, The minimum area of convex lattice n-gons

Cai, On the minimum area of convex lattice polygons

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

See Barany & Norihide for asymptotics.

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.

Cf. A070911.

Sequence in context: A049647 A046538 A107432 this_sequence A011981 A109544 A036605

Adjacent sequences: A063981 A063982 A063983 this_sequence A063985 A063986 A063987

nice,nonn

Pierre Bornsztein (pbornszt(AT)club-internet.fr), Sep 06 200; May 20, 2002

Additional comments from S. R. Finch, Dec 06 2003