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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.
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%I #30 Jun 23 2020 09:11:19

%S 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,

%T 70,70,45,29,29,45,70,70,126,140,105,72,64,72,105,140,126,210,252,210,

%U 157,129,129,157,210,252,210,330,420,378,302,248,234,248,302,378,420,330

%N 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.

%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, 1, 5, 15, 35, 70, 126, 210, 330, 495, ...

%e 0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, ...

%e 0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, ...

%e 1, 2, 3, 10, 29, 72, 157, 302, 531, 874, 1361, 2028, ...

%e 5, 10, 15, 29, 64, 129, 248, 442, 747, 1196, 1825, 2679, ...

%e 15, 30, 45, 72, 129, 234, 405, 666, 1065, 1638, 2439, 3510, ...

%e 35, 70, 105, 157, 248, 405, 660, 1020, 1545, 2276, 3283, 4605, ...

%e 70, 140, 210, 302, 442, 666, 1020, 1524, 2220, 3154, 4412, 6030, ...

%e 126, 252, 378, 531, 747, 1065, 1545, 2220, 3156, 4362, 5940, 7923, ...

%e 210, 420, 630, 874, 1196, 1638, 2276, 3154, 4362, 5928, 7914, 10350, ...

%e ...

%e The initial antidiagonals are:

%e 0

%e 0, 0

%e 0, 0, 0

%e 1, 0, 0, 1

%e 5, 2, 0, 2, 5

%e 15, 10, 3, 3, 10, 15

%e 35, 30, 15, 10, 15, 30, 35

%e 70, 70, 45, 29, 29, 45, 70, 70

%e 126, 140, 105, 72, 64, 72, 105, 140, 126

%e 210, 252, 210, 157, 129, 129, 157, 210, 252, 210

%e 330, 420, 378, 302, 248, 234, 248, 302, 378, 420, 330

%e ...

%Y The main diagonal is A178256.

%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 1,11

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