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

Author

Martin Renner, Apr 21 2011

Keywords

• nonn

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.

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