A334709 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 strictly inside the triangle.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 48, 48, 0, 0, 0, 0, 144, 240, 144, 0, 0, 0, 0, 348, 716, 716, 348, 0, 0, 0, 0, 700, 1712, 2100, 1712, 700, 0, 0, 0, 0, 1280, 3404, 4984, 4984, 3404, 1280, 0, 0, 0, 0, 2144, 6176, 9900, 11604, 9900, 6176, 2144, 0, 0, 0, 0, 3400, 10336, 17936, 22936, 22936, 17936, 10336, 3400, 0, 0

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, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 8, 48, 144, 348, 700, 1280, 2144, 3400, 5120, 7440, ...
0, 0, 48, 240, 716, 1712, 3404, 6176, 10336, 16288, 24480, 35504, ...
0, 0, 144, 716, 2100, 4984, 9900, 17936, 29924, 47080, 70700, 102460, ...
0, 0, 348, 1712, 4984, 11604, 22936, 41372, 68844, 108132, 161964, 234228, ...
0, 0, 700, 3404, 9900, 22936, 45184, 81320, 135192, 212152, 317492, 458812, ...
0, 0, 1280, 6176, 17936, 41372, 81320, 145648, 241544, 378400, 565636, 816520, ...
0, 0, 2144, 10336, 29924, 68844, 135192, 241544, 399656, 625232, 933808, 1346928, ...
0, 0, 3400, 16288, 47080, 108132, 212152, 378400, 625232, 976552, 1457172, 2100112, ...
...
The initial antidiagonals are:
0,
0, 0,
0, 0, 0,
0, 0, 0, 0,
0, 0, 8, 0, 0,
0, 0, 48, 48, 0, 0,
0, 0, 144, 240, 144, 0, 0,
0, 0, 348, 716, 716, 348, 0, 0,
0, 0, 700, 1712, 2100, 1712, 700, 0, 0,
0, 0, 1280, 3404, 4984, 4984, 3404, 1280, 0, 0,
0, 0, 2144, 6176, 9900, 11604, 9900, 6176, 2144, 0, 0,
0, 0, 3400, 10336, 17936, 22936, 22936, 17936, 10336, 3400, 0, 0,
...
		

Links

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

Crossrefs

The main diagonal is A334712.

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.