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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 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.)