A189413 Number of convex quadrilaterals on an n X n grid (or geoboard).

0, 1, 70, 1038, 7398, 35727, 130768, 400116, 1062016, 2531001, 5529310, 11272710, 21639022, 39559591, 69283632, 116910052, 190977408, 303286461, 469431366, 710400658, 1053055398, 1532253131, 2192246528, 3088876728, 4290532688, 5882825641, 7969711934, 10677299074, 14156978846, 18591603883, 24195121104

Offset

1,3

Comments

If four points are chosen at random from an n X n grid, the probability that they form a convex quadrilateral approaches 25/36 as n increases, by Sylvester's Four-Point Theorem (see the link). Thanks to Ed Pegg Jr for this comment. - N. J. A. Sloane, Jun 15 2020

Links

Tom Duff, Table of n, a(n) for n = 1..192

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192.

Nathaniel Johnston, C program for computing terms.

Eric Weisstein's World of Mathematics, Convex Polygon.

Eric Weisstein's World of Mathematics, Quadrilateral.

Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Crossrefs

Cf. A175383, A178256, A181944, A189345, A189412, A189414.

This is the main diagonal of A334711.

Sequence in context: A061170 A125114 A281649 * A362054 A229735 A254472

Adjacent sequences: A189410 A189411 A189412 * A189414 A189415 A189416

Keyword

nonn

Author

Martin Renner, Apr 21 2011

Extensions

a(6) - a(22) from Nathaniel Johnston, Apr 25 2011

Terms beyond a(22) from Tom Duff. - N. J. A. Sloane, Jun 23 2020

Status

approved