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

1,5

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]

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.

Example

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, ...

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 *)

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.

Sequence in context: A046617 A131063 A081579 * A146772 A202868 A202877

Adjacent sequences: A295704 A295705 A295706 * A295708 A295709 A295710

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 26 2017

Status

approved