A018809 -id:A018809 - cp's OEIS frontend

A119437 Table T(n,k) = number of lines through exactly k points of an n X n grid of points.

6, 12, 8, 48, 4, 10, 108, 16, 4, 12, 248, 36, 4, 4, 14, 428, 64, 20, 4, 4, 16, 764, 100, 44, 4, 4, 4, 18, 1196, 204, 36, 24, 4, 4, 4, 20, 1900, 252, 64, 52, 4, 4, 4, 4, 22, 2668, 396, 124, 40, 28, 4, 4, 4, 4, 24, 3824, 572, 200, 20, 60, 4, 4, 4, 4, 4, 26, 5244, 780, 236, 76, 44, 32

Offset: 2

Franklin T. Adams-Watters, May 19 2006

			From _Seiichi Manyama_, Nov 26 2017: (Start)
The table starts:
  n\k|   2    3   4   5   6   7   8
  ---+------------------------------
   2 |   6;
   3 |  12,   8;
   4 |  48,   4, 10;
   5 | 108   16,  4, 12;
   6 | 248,  36,  4,  4, 14;
   7 | 428,  64, 20,  4,  4, 16;
   8 | 764, 100, 44,  4,  4,  4, 18; (End)

Seiichi Manyama, Rows n = 2..141, flattened
S. Mustonen, On lines and their intersection points in a rectangular grid of points [From _Seppo Mustonen_, Apr 18 2009]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]

Row sums A018808; columns A018809-A018817. See A119439 for another version.

T(n,k) = 1/2 (f(n, k+1) - 2 f(n, k) + f(n, k-1)) where f(n, k) = Sum_{-n < kx < n, -n < ky < n, gcd(x, y)=1} (n - |kx|)*(n - |ky|). [Seppo Mustonen, Apr 18 2009]

An incorrect formula removed by Seppo Mustonen, Apr 25 2009

A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).

Original entry on oeis.org

0, 0, 0, 0, 10, 16, 22, 36, 54, 66, 92, 122, 156, 196, 240, 288, 366, 426, 490, 590, 698, 780, 904, 1036, 1176, 1326, 1484, 1650, 1874, 2060, 2254, 2512, 2782, 3006, 3300, 3606, 3924, 4256, 4600, 4956, 5398, 5782, 6178, 6666, 7170, 7608, 8144

Offset: 0

Sean A. Irvine, Mar 23 2011

In contrast, the number of distinct lines passing through 4 or more points in an n X n grid is given by 0, 0, 0, 10, 16, 22, 44, 74, 92, 154, 232, 326, 436, 562, 704, 998, 1268,.. = A018808(n) -A018809(n) -A018810(n) = A225606(n) -A018810(n). - David W. Wilson, Aug 05 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 2, 0, -1, -4, -1, 0, 2, 2, 0, 0, -1).

Cf. A178465, A187062.

CoefficientList[ Series[ 2x^4 (5 + 8x + 11x^2 + 8x^3 + x^4 - 5x^5 - 7x^6 - x^7 + 6x^8 + 10x^9 + 6x^10 - x^11 - 2x^12 - 3x^13)/((-1 + x)^4 (1 + x)^2 (1 + x^2)^2 (1 + x + x^2)^2), {x, 0, 43}], x] (* or *) LinearRecurrence[{0, 0, 2, 2, 0, -1, -4, -1, 0, 2, 2, 0, 0, -1}, {10, 16, 22, 36, 54, 66, 92, 122, 156, 196, 240, 288, 366, 426}, 40] (* Robert G. Wilson v, Feb 17 2014 *)

Definition replaced with Colin Barker's g.f. by R. J. Mathar, Aug 06 2013

Offset changed from 1 to 0 and a(0)=0 added by Vincenzo Librandi, Feb 19 2014

A225606 Number of distinct lines passing through 3 or more points in an n X n grid.

Original entry on oeis.org

0, 0, 0, 8, 14, 32, 58, 108, 174, 296, 406, 628, 898, 1216, 1582, 2188, 2754, 3528, 4398, 5524, 6778, 8336, 9778, 11812, 14038, 16456, 19066, 22540, 25954, 29968, 34270, 39116, 44282, 50312, 56026, 63196, 70798, 78984

Offset: 0

R. J. Mathar, Aug 06 2013

nonn

P. Haukkanen, J. K. Merikoski, Some formulas for numbers of line segments and lines in a rectangular grid, arXiv:1108.1041
R. J. Mathar, Maple program implementing A225606, A018808 and sequ. in comment of A187397

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}];
a[n_] := (f[n, 2] - f[n, 3])/2;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 30 2018, after Seppo Mustonen in A018808 *)

a(n) = A018808(n) - A018809(n) = A018810(n) + A018811(n) + A018812(n) + A018813(n)+....

A119439 Triangle T(n,k) = number of sets of m points determined by the intersection of a line with an n X n grid of points.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 1, 9, 12, 8, 1, 16, 48, 4, 10, 1, 25, 108, 16, 4, 12, 1, 36, 248, 36, 4, 4, 14, 1, 49, 428, 64, 20, 4, 4, 16, 1, 64, 764, 100, 44, 4, 4, 4, 18, 1, 81, 1196, 204, 36, 24, 4, 4, 4, 20, 1, 100, 1900, 252, 64, 52, 4, 4, 4, 4, 22, 1, 121, 2668, 396, 124, 40, 28, 4, 4, 4, 4

Offset: 0

Franklin T. Adams-Watters, May 19 2006

Each singleton point is determined by all but finitely many of the family of lines passing through that point and the empty set is determined by any randomly positioned line.

			The table starts:
1,
1,1,
1,4,6,
1,9,12,8,
1,16,48,4,10,

Row sums A119438; columns A000290, A018809-A018817. See A119437 for another version.

T(n,0) = 1, T(n,1) = n^2, T(n,k) = A119437(n,k) for k>1.

cp's OEIS Frontend

A119437 Table T(n,k) = number of lines through exactly k points of an n X n grid of points.

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A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).

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A225606 Number of distinct lines passing through 3 or more points in an n X n grid.

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Programs

Mathematica

Formula

A119439 Triangle T(n,k) = number of sets of m points determined by the intersection of a line with an n X n grid of points.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

cp's OEIS Frontend

A119437 Table T(n,k) = number of lines through exactly k points of an n X n grid of points.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A187397 Expansion of -2*x^4 *(3*x^13 +2*x^12 +x^11 -6*x^10 -10*x^9 -6*x^8 +x^7 +7*x^6 +5*x^5 -x^4 -8*x^3 -11*x^2 -8*x -5) / ((x -1)^4 *(x +1)^2 *(x^2 +1)^2 *(x^2 +x +1)^2).

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Programs

Mathematica

Extensions

A225606 Number of distinct lines passing through 3 or more points in an n X n grid.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A119439 Triangle T(n,k) = number of sets of m points determined by the intersection of a line with an n X n grid of points.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).