%I #17 Jun 23 2020 09:11:51

%S 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,

%T 348,716,716,348,0,0,0,0,700,1712,2100,1712,700,0,0,0,0,1280,3404,

%U 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

%N 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.

%C Computed by _Tom Duff_, Jun 15 2020

%H Tom Duff, <a href="/A334708/a334708_3.txt">Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192</a>

%e The initial rows of the array are:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 0, 8, 48, 144, 348, 700, 1280, 2144, 3400, 5120, 7440, ...

%e 0, 0, 48, 240, 716, 1712, 3404, 6176, 10336, 16288, 24480, 35504, ...

%e 0, 0, 144, 716, 2100, 4984, 9900, 17936, 29924, 47080, 70700, 102460, ...

%e 0, 0, 348, 1712, 4984, 11604, 22936, 41372, 68844, 108132, 161964, 234228, ...

%e 0, 0, 700, 3404, 9900, 22936, 45184, 81320, 135192, 212152, 317492, 458812, ...

%e 0, 0, 1280, 6176, 17936, 41372, 81320, 145648, 241544, 378400, 565636, 816520, ...

%e 0, 0, 2144, 10336, 29924, 68844, 135192, 241544, 399656, 625232, 933808, 1346928, ...

%e 0, 0, 3400, 16288, 47080, 108132, 212152, 378400, 625232, 976552, 1457172, 2100112, ...

%e ...

%e The initial antidiagonals are:

%e 0,

%e 0, 0,

%e 0, 0, 0,

%e 0, 0, 0, 0,

%e 0, 0, 8, 0, 0,

%e 0, 0, 48, 48, 0, 0,

%e 0, 0, 144, 240, 144, 0, 0,

%e 0, 0, 348, 716, 716, 348, 0, 0,

%e 0, 0, 700, 1712, 2100, 1712, 700, 0, 0,

%e 0, 0, 1280, 3404, 4984, 4984, 3404, 1280, 0, 0,

%e 0, 0, 2144, 6176, 9900, 11604, 9900, 6176, 2144, 0, 0,

%e 0, 0, 3400, 10336, 17936, 22936, 22936, 17936, 10336, 3400, 0, 0,

%e ...

%Y The main diagonal is A334712.

%Y Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.

%Y For three points there are just two possible arrangements: see A334704 and A334705.

%K nonn,tabl

%O 1,13

%A _N. J. A. Sloane_, Jun 15 2020