A334708 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four collinear points from an n X k grid.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 5, 2, 0, 2, 5, 15, 10, 3, 3, 10, 15, 35, 30, 15, 10, 15, 30, 35, 70, 70, 45, 29, 29, 45, 70, 70, 126, 140, 105, 72, 64, 72, 105, 140, 126, 210, 252, 210, 157, 129, 129, 157, 210, 252, 210, 330, 420, 378, 302, 248, 234, 248, 302, 378, 420, 330

Offset: 1

Author

N. J. A. Sloane, Jun 15 2020

Keywords

• nonn
• tabl

Comments

Computed by Tom Duff, Jun 15 2020

Examples

			The initial rows of the array are:
0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, ...
0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, ...
0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, ...
1, 2, 3, 10, 29, 72, 157, 302, 531, 874, 1361, 2028, ...
5, 10, 15, 29, 64, 129, 248, 442, 747, 1196, 1825, 2679, ...
15, 30, 45, 72, 129, 234, 405, 666, 1065, 1638, 2439, 3510, ...
35, 70, 105, 157, 248, 405, 660, 1020, 1545, 2276, 3283, 4605, ...
70, 140, 210, 302, 442, 666, 1020, 1524, 2220, 3154, 4412, 6030, ...
126, 252, 378, 531, 747, 1065, 1545, 2220, 3156, 4362, 5940, 7923, ...
210, 420, 630, 874, 1196, 1638, 2276, 3154, 4362, 5928, 7914, 10350, ...
...
The initial antidiagonals are:
0
0, 0
0, 0, 0
1, 0, 0, 1
5, 2, 0, 2, 5
15, 10, 3, 3, 10, 15
35, 30, 15, 10, 15, 30, 35
70, 70, 45, 29, 29, 45, 70, 70
126, 140, 105, 72, 64, 72, 105, 140, 126
210, 252, 210, 157, 129, 129, 157, 210, 252, 210
330, 420, 378, 302, 248, 234, 248, 302, 378, 420, 330
...
		

Links

• Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

Crossrefs

The main diagonal is A178256.

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.