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

Author

David W. Wilson

Keywords

• nonn

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]

Programs

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 *)

Formula

a(n) = (1/2) * (f(n, 3) - 2 f(n, 2) + f(n, 1)) where f(n, k) = Sum ((n - |kx|)(n - |ky|)); -nSeppo Mustonen, Apr 18 2009]

Extensions

An incorrect formula removed by Seppo Mustonen, Apr 25 2009