

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
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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 FourPoint 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 FourPoint Problem, From MathWorldA 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



