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

1,8

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.

Links

Table of n, a(n) for n=1..66.

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.

Example

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,
...

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.

Sequence in context: A165398 A329716 A021105 * A176537 A019891 A176536

Adjacent sequences: A334708 A334709 A334710 * A334712 A334713 A334714

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 15 2020

Status

approved