A018809 Number of lines through exactly 2 points of an n X n grid of points.

0, 0, 6, 12, 48, 108, 248, 428, 764, 1196, 1900, 2668, 3824, 5244, 7248, 9380, 12192, 15372, 19528, 24020, 29732, 36052, 43808, 51836, 61636, 72492, 85308, 98492, 114012, 130668, 150196, 170828, 194768, 220276, 249452, 279284, 312572, 348036

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

S. Mustonen, On lines and their intersection points in a rectangular grid of points [From Seppo Mustonen, Apr 18 2009]

Formula

a(n) = (1/2) * (f(n, 3) - 2 f(n, 2) + f(n, 1)) where f(n, k) = Sum ((n - |kx|)(n - |ky|)); -n<kx<n, -n<ky<n, (x, y)=1. [Seppo Mustonen, Apr 18 2009]

Mathematica

a[n_] := 1/2 (f[n, 3] - 2 f[n, 2] + f[n, 1]);
f[n_, k_] := Sum[x = kx/k; y = ky/k; If[IntegerQ[x] && IntegerQ[y] && CoprimeQ[x, y], (n - Abs[kx])(n - Abs[ky]), 0], {kx, -n + 1, n - 1}, {ky, -n + 1, n - 1}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 30 2018 *)

Crossrefs

Sequence in context: A106692 A032470 A342302 * A226882 A214903 A256584

Adjacent sequences: A018806 A018807 A018808 * A018810 A018811 A018812

Keyword

nonn

Author

David W. Wilson

Extensions

An incorrect formula removed by Seppo Mustonen, Apr 25 2009

Status

approved