cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A000769 No-3-in-line problem: number of inequivalent ways of placing 2n points on an n X n grid so that no 3 are in a line.

Original entry on oeis.org

0, 1, 1, 4, 5, 11, 22, 57, 51, 156, 158, 566, 499, 1366, 3978, 5900, 7094, 19204
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

This means no three points on any line, not just lines in the X or Y directions.
A000755 gives the total number of solutions (as opposed to the number of equivalence classes).
It is conjectured that a(n)=0 for all sufficiently large n.
Flammenkamp's web site reports that at least one solution is known for all n <= 46 and n=48, 50, 52.
From R. K. Guy, Oct 22 2004: (Start)
I got the no-three-in-line problem from Heilbronn over 50 years ago. See Section F4 in UPINT.
In Canad. Math. Bull. 11 (1968) 527-531, MR 39 #129, Guy & Kelly conjecture that, for large n, at most (c + eps)*n points can be selected, where 3*c^3 = 2*Pi^2, i.e., c ~ 1.87.
As recently as last March, Gabor Ellmann pointed out an error in our heuristic reasoning, which, when corrected, gives 3*c^2 = Pi^2, or c ~ 1.813799. (End)

Examples

			a(3) = 1:
  X X o
  X o X
  o X X

References

  • M. A. Adena, D. A. Holton and P. A. Kelly, Some thoughts on the no-three-in-line problem, pp. 6-17 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
  • D. B. Anderson, Journal of Combinatorial Theory Series A, V.27/1979 pp. 365 - 366.
  • D. Craggs and R. Hughes-Jones, Journal of Combinatorial Theory Series A, V. 20/1976 pp. 363-364.
  • H. E. Dudeney, Amusements in Mathematics, Nelson, Edinburgh 1917, pp. 94, 222.
  • M. Gardner, Scientific American V236 / March 1977, pp. 139-140.
  • M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 69.
  • R. K. Guy, Unsolved combinatorial problems, pp. 121-127 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
  • R. K. Guy, Unsolved Problems Number Theory, Section F4.
  • R. K. Guy and P. A. Kelly, The No-Three-Line Problem. Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. Condensed version in Canad. Math. Bull. Vol. 11, pp. 527-531, 1968.
  • R. R. Hall, T. H. Jackson, A. Sudberry and K. Wild, Journal of Combinatorial Theory Series A, V.18/1975 pp. 336-341.
  • H. Harborth, P. Oertel and T. Prellberg, Discrete Math. V73/1988 pp. 89-90.
  • T. Kløve, Journal of Combinatorial Theory Series A, V.24/1978 pp. 126-127.
  • T. Kløve, Journal of Combinatorial Theory Series A, V.26/1979 pp. 82-83.
  • K. F. Roth, Journal London Math. Society V.26 / 1951, p. 204.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000755, A000938, A037185, A037186, A037187, A037188, A037189, A047840, A212807, A235453.
See A272651 for the maximal number of no-3-in-line points on an n X n grid, and A277433 for minimal saturated.
Cf. A194136 (triangular grid), A280537 (3D grid, no 4 in plane).

Extensions

a(17) and a(18) from Benjamin Chaffin, Apr 05 2006
Minor edits from N. J. A. Sloane, May 25 2010
Edited by N. J. A. Sloane, Mar 19 2013 at the suggestion of Dominique Bernardi

A279453 Triangle read by rows: T(n, k) is the number of nonequivalent ways to place k points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 14, 17, 9, 2, 1, 3, 21, 73, 202, 306, 285, 115, 20, 1, 6, 49, 301, 1397, 4361, 9110, 11810, 8679, 2929, 288, 1, 6, 93, 890, 6582, 34059, 126396, 326190, 568134, 624875, 390426, 111798, 8791, 1, 10, 171, 2321, 24185, 185181, 1055025
Offset: 1

Views

Author

Heinrich Ludwig, Dec 17 2016

Keywords

Comments

Length of n-th row is A272651(n) + 1, where A272651(n) is the maximal number of points that can be placed under the condition mentioned.
Rotations and reflections of placements are not counted. If they are to be counted, see A279445.
For condition "no more than 2 points on a straight line at any angle", see A235453.

Examples

			The table begins with T(1, 0):
1 1
1 1  2   1    1
1 3  8  14   17    9    2
1 3 21  73  202  306  285   115   20
1 6 49 301 1397 4361 9110 11810 8679 2929 288
...
T(4, 3) = 73 because there are 73 nonequivalent ways to place 3 points on a 4 X 4 square grid so that no more than 2 points are on a vertical or horizontal straight line.

Links

Crossrefs

Row sums give A279454.
Columns 2..8: A008805, A014409, A279454, A279455, A279456, A279457, A279458.
Diagonal T(n, n) is A279452.
Cf. A279445, A235453.

A235454 Number of non-equivalent (mod D_4) ways to arrange 3 points on an n X n square grid so that they are not collinear.

Original entry on oeis.org

1, 13, 70, 290, 867, 2266, 5068, 10475, 19764, 35406, 59817, 97375, 152154
Offset: 2

Views

Author

Heinrich Ludwig, Jan 12 2014

Keywords

Comments

Column 3 of A235453.
Also number of non-equivalent triangles in an n X n grid.
Without the restriction "non-equivalent (mod D_4)" the numbers are given by A045996, n >= 2.

Examples

			There are a(3) = 13 non-equivalent ways to place 3 points on a 3 X 3 grid:
  . . .   . . .   X . .   . X .   . . .   . X .   . X .
  X . .   X . .   . . .   . . .   . X .   X . X   X X .
  X X .   X . X   X . X   X . X   X . X   . . .   . . .
-
  . . .   . . .   . . .   . X .   . X .   . . X
  X . X   . X X   X X .   . . X   X . .   X . .
  X . .   X . .   X . .   X . .   X . .   X . .

Crossrefs

Cf. A235453, A045996, A235455 (4 points), A235456 (5 points), A235457 (6 points), A235458 (7 points).

Extensions

a(13), a(14) from Heinrich Ludwig, Nov 16 2016

A235455 Number of non-equivalent (mod D_4) ways to arrange 4 points on an n X n square grid so that no three points are collinear.

Original entry on oeis.org

1, 15, 181, 1253, 6044, 22302, 68661, 183645, 439578, 964938, 1974128, 3801457, 6966581
Offset: 2

Views

Author

Heinrich Ludwig, Jan 12 2014

Keywords

Comments

Column 4 of A235453.
Also number of non-equivalent complete quadrangles on an n X n grid.
Without the restriction "non-equivalent (mod D_4)" the numbers are given by A175383, n >= 2.

Examples

			There are a(3) = 15 non-equivalent ways to place 4 points (X) on a 3 X 3 grid. Examples are:
  X . X    . X .    X X .
  . . .    X . X    X . .
  X . X    . X .    . . X

Crossrefs

Cf. A235453, A045996, A235454 (3 points), A235456 (5 points), A235457 (6 points), A235458 (7 points)

Extensions

a(13), a(14) from Heinrich Ludwig, Nov 16 2016

A235456 Number of non-equivalent (mod D_4) ways to arrange 5 points on an n X n square grid so that no three points are collinear.

Original entry on oeis.org

5, 217, 3192, 27041, 149217, 650566, 2317137, 7124316, 19459757, 48617666, 111797647, 241575473
Offset: 3

Views

Author

Heinrich Ludwig, Jan 12 2014

Keywords

Comments

Column 5 of A235453.
Without the restriction "non-equivalent (mod D_4)" the numbers are given by A194190, n >= 3.

Examples

			There are a(3) = 5 non-equivalent ways to place 5 points (X) on a 3 X 3 grid:
  X . X    X . X    . X X    X X .    X . X
  X . X    X . .    X . X    X . .    X X .
  . X .    . X X    . X .    . X X    . X .

Crossrefs

Cf. A235453, A194190, A235454 (3 points), A235455 (4 points), A235457 (6 points), A235458 (7 points)

Extensions

a(13), a(14) from Heinrich Ludwig, Nov 16 2016

A235457 Number of non-equivalent (mod D_4) ways to arrange 6 points on an n X n square grid so that no three points are collinear.

Original entry on oeis.org

1, 142, 4699, 77970, 672506, 4338248, 21167201, 85351595, 294664274, 911848844, 2528561187, 6501165477
Offset: 3

Views

Author

Heinrich Ludwig, Jan 12 2014

Keywords

Comments

Column 6 of A235453.
Without the restriction "non-equivalent (mod D_4)" the numbers are given by A194191, n >= 3.

Examples

			There is a(3) = 1 way to place 6 points (X) on a 3 X 3 grid (without rotations and reflections):
   . X X
   X . X
   X X .

Crossrefs

Cf. A235453, A194191, A235454 (3 points), A235455 (4 points), A235456 (5 points), A235458 (7 points).

Extensions

a(13), a(14) from Heinrich Ludwig, Nov 16 2016

A235458 Number of non-equivalent (mod D_4) ways to arrange 7 points on an n X n square grid so that no three points are collinear.

Original entry on oeis.org

28, 3385, 134353, 1958674, 19929645, 138586349, 753795278, 3356614240, 13108210508, 44374441652, 137349454120
Offset: 4

Views

Author

Heinrich Ludwig, Jan 12 2014

Keywords

Comments

Column 7 of A235453.
Without the restriction "non-equivalent (mod D_4)" the numbers are given by A194192, n >= 4.

Examples

			There are a(4) = 28 non-equivalent ways to place 7 points (X) on a 4 X 4 grid. Example:
   . X X .
   . . . X
   X . . X
   X X . .

Crossrefs

Cf. A235453, A194192, A235454 (3 points), A235455 (4 points), A235456 (5 points), A235457 (6 points).

Extensions

a(13), a(14) from Heinrich Ludwig, Nov 16 2016
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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