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

1,13

Comments

Computed by Tom Duff, Jun 15 2020

Links

Table of n, a(n) for n=1..78.

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, 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,
...

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.

Sequence in context: A374087 A325737 A272625 * A220667 A317445 A199619

Adjacent sequences: A334706 A334707 A334708 * A334710 A334711 A334712

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 15 2020

Status

approved