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

0,3

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]

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

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 *)

Crossrefs

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

A288187 is the main entry for these graphs.

Cf. A331780.

Sequence in context: A027984 A342313 A309294 * A027107 A247307 A279215

Adjacent sequences: A018805 A018806 A018807 * A018809 A018810 A018811

Keyword

nonn,nice

Author

David W. Wilson

Status

approved