A334710 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 three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 32, 48, 32, 0, 0, 100, 168, 168, 100, 0, 0, 240, 456, 532, 456, 240, 0, 0, 490, 990, 1312, 1312, 990, 490, 0, 0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0, 0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0, 0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0

Offset

1,8

Comments

Computed by Tom Duff, Jun 15 2020

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

Example

The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 6, 32, 100, 240, 490, 896, 1512, 2400, 3630, 5280, ...
0, 6, 48, 168, 456, 990, 1920, 3360, 5520, 8550, 12720, 18216, ...
0, 32, 168, 532, 1312, 2652, 4972, 8420, 13452, 20480, 29980, 42288, ...
0, 100, 456, 1312, 3088, 5964, 10816, 17768, 27840, 41652, 60040, 83448, ...
0, 240, 990, 2652, 5964, 11340, 20142, 32436, 50004, 73704, 105282, 144936, ...
0, 490, 1920, 4972, 10816, 20142, 35264, 55916, 84960, 123690, 174976, 238512, ...
0, 896, 3360, 8420, 17768, 32436, 55916, 88088, 132708, 191588, 268972, 363876, ...
0, 1512, 5520, 13452, 27840, 50004, 84960, 132708, 198912, 285312, 397968, 534888, ...
0, 2400, 8550, 20480, 41652, 73704, 123690, 191588, 285312, 407744, 566046, 757008, ...
...
The initial antidiagonals are:
0
0, 0
0, 0, 0
0, 6, 6, 0
0, 32, 48, 32, 0
0, 100, 168, 168, 100, 0
0, 240, 456, 532, 456, 240, 0
0, 490, 990, 1312, 1312, 990, 490, 0
0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0
0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0
0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0
0, 3630, 8550, 13452, 17768, 20142, 20142, 17768, 13452, 8550, 3630, 0
...

Crossrefs

The main diagonal is A334713.

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: A055667 A281080 A283347 * A283203 A155742 A364406

Adjacent sequences: A334707 A334708 A334709 * A334711 A334712 A334713

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 15 2020

Status

approved