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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.
7
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.
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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 15 2020
STATUS
approved