A295707 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.

0, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 18, 20, 18, 1, 1, 27, 35, 35, 27, 1, 1, 38, 52, 62, 52, 38, 1, 1, 51, 75, 93, 93, 75, 51, 1, 1, 66, 100, 136, 140, 136, 100, 66, 1, 1, 83, 131, 181, 207, 207, 181, 131, 83, 1, 1, 102, 164, 238, 274, 306, 274, 238, 164, 102, 1

Offset: 1

Author

Seiichi Manyama, Nov 26 2017

Keywords

• nonn
• tabl

Examples

			Square array begins:
   0,  1,  1,   1,   1, ...
   1,  6, 11,  18,  27, ...
   1, 11, 20,  35,  52, ...
   1, 18, 35,  62,  93, ...
   1, 27, 52,  93, 140, ...
   1, 38, 75, 136, 207, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..140, flattened
• Seppo Mustonen, On lines and their intersection points in a rectangular grid of points
• Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]

Crossrefs

Columns k=2..10 give A160842, A160843, A160844, A160845, A160846, A160847, A160848, A160849, A160850.

Main diagonal gives A018808. Reading up to the diagonal gives A107348.

Programs

Mathematica

A[n_, k_] := (1/2)(f[n, k, 1] - f[n, k, 2]);
f[n_, k_, m_] := Sum[If[GCD[mx/m, my/m] == 1, (n - Abs[mx])(k - Abs[my]), 0], {mx, -n, n}, {my, -k, k}];
Table[A[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2023 *)

Formula

A(n,k) = (1/2) * (f(n,k,1) - f(n,k,2)), where f(n,k,m) = Sum ((n-|m*x|)*(k-|m*y|)); -n < m*x < n, -k < m*y < k, (x,y)=1.