A334711 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that they form a convex quadrilateral.

0, 0, 0, 0, 1, 0, 0, 9, 9, 0, 0, 36, 70, 36, 0, 0, 100, 276, 276, 100, 0, 0, 225, 750, 1038, 750, 225, 0, 0, 441, 1677, 2788, 2788, 1677, 441, 0, 0, 784, 3260, 6190, 7398, 6190, 3260, 784, 0, 0, 1296, 5776, 11942, 16328, 16328, 11942, 5776, 1296, 0, 0, 2025, 9508, 21062, 31396, 35727, 31396, 21062, 9508, 2025, 0

Offset: 1

Author

N. J. A. Sloane, Jun 15 2020

Keywords

• nonn
• tabl

Comments

Computed by Tom Duff, Jun 15 2020

For the limiting probability that the four points form a convex quadrilateral when n and k are large, see the link to Sylvester's Four-Point Problem. Thanks to Ed Pegg Jr for this comment.

Examples

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, ...
0, 9, 70, 276, 750, 1677, 3260, 5776, 9508, 14825, 22090, 31764, ...
0, 36, 276, 1038, 2788, 6190, 11942, 21062, 34586, 53748, 79930, 114760, ...
0, 100, 750, 2788, 7398, 16328, 31396, 55244, 90484, 140372, 208490, 299048, ...
0, 225, 1677, 6190, 16328, 35727, 68447, 120106, 196338, 304161, 451035, 646116, ...
0, 441, 3260, 11942, 31396, 68447, 130768, 229034, 373968, 578777, 857524, 1227572, ...
0, 784, 5776, 21062, 55244, 120106, 229034, 400116, 652318, 1008438, 1492870, 2135534, ...
0, 1296, 9508, 34586, 90484, 196338, 373968, 652318, 1062016, 1640284, 2426660, 3469356, ...
0, 2025, 14825, 53748, 140372, 304161, 578777, 1008438, 1640284, 2531001, 3742053, 5347100, ...
...
The initial antidiagonals are:
0,
0, 0,
0, 1, 0,
0, 9, 9, 0,
0, 36, 70, 36, 0,
0, 100, 276, 276, 100, 0,
0, 225, 750, 1038, 750, 225, 0,
0, 441, 1677, 2788, 2788, 1677, 441, 0,
0, 784, 3260, 6190, 7398, 6190, 3260, 784, 0,
0, 1296, 5776, 11942, 16328, 16328, 11942, 5776, 1296, 0,
0, 2025, 9508, 21062, 31396, 35727, 31396, 21062, 9508, 2025, 0,
0, 3025, 14825, 34586, 55244, 68447, 68447, 55244, 34586, 14825, 3025, 0,
...
		

Links

• Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192
• Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Crossrefs

The main diagonal is A189413.

Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.

For three points there are just two possible arrangements: see A334704 and A334705.