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A189413 Number of convex quadrilaterals on an n X n grid (or geoboard). 10
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 (list; graph; refs; listen; history; text; internal format)
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

Weisstein, Eric W. Sylvester's Four-Point Problem, From MathWorld--A Wolfram Web Resource.

Weisstein, Eric W.: MathWorld -- Convex Polygon.

Weisstein, Eric W.: MathWorld -- Quadrilateral.

CROSSREFS

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

This is the main diagonal of A334711.

Sequence in context: A061170 A125114 A281649 * A229735 A254472 A353896

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

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)