A018808 Number of lines through at least 2 points of an n X n grid of points.

0, 0, 6, 20, 62, 140, 306, 536, 938, 1492, 2306, 3296, 4722, 6460, 8830, 11568, 14946, 18900, 23926, 29544, 36510, 44388, 53586, 63648, 75674, 88948, 104374, 121032, 139966, 160636, 184466, 209944, 239050, 270588, 305478, 342480, 383370, 427020

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• nice

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
• M. A. Alekseyev, M. Basova, N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM J. Disc. Math. 29(1), 2015, pp. 157-165.
• A.-M. Ernvall-Hytonen, K. Matomaki, P. Haukkanen, J. K. Merikoski, Formulas for the number of gridlines, Monatsh. f. Mathem. 164 (2) (2011) 157-170
• P. Haukkanen, J. K. Merikoski, Some formulas for numbers of line segments and lines in a rectangular grid, arXiv:1108.1041 [math.CO], 2011.
• S. 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

Cf. A222267 (lines defined by n X n X n grid of points).

A288187 is the main entry for these graphs.

Cf. A331780.

Programs

Mathematica

L[0]=0; L1[1]=0; R1[1]=0;
L[n_]:=L[n]=2*L1[n]-L[n-1]+R1[n]
L1[n_]:=L1[n]=2*L[n-1]-L1[n-1]+R2[n]
R1[n_]:=R1[n]=R1[n-1]+4*(EulerPhi[n-1]-e[n])
e[n_]:=If[Mod[n,2]==0,0,EulerPhi[(n-1)/2]]
R2[n_]:= If[Mod[n,2]==0,(n-1)*EulerPhi[n-1], If[Mod[n,4]==1,(n-1)*EulerPhi[n-1]/2,0]]
Table[L[n],{n,0,37}] (* Seppo Mustonen, Apr 25 2009 *)

Formula

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

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

a(0) = L(0,1) = R1(0) = 0, a(n) = L(n,n) = 2L(n-1,n) - L(n-1,n-1) + R1(n), L(n-1,n) = 2L(n-1,n-1) - L(n-2,n-1) + R2(n), R1(n) = R1(n-1) + 4(phi(n-1) - e(n)), e(n)=0, n even, e(n) = phi((n-1)/2), n odd, R2(n) = (n-1)phi(n-1), n even, R2(n)=(n-1)phi(n-1)/2, n=1 mod 4, R2(n)=0, n=3 mod 4. - Seppo Mustonen, Apr 25 2009

a(n) = 2 * A331780(n). - Alois P. Heinz, Jun 05 2023