A019654 -id:A019654 - cp's OEIS frontend

A051567 Consider problem of placing N queens on an n X n board so that each queen attacks precisely k others. Here k=1 and sequence gives number of inequivalent solutions when N is equal to the upper bound 2*floor(2n/3).

Original entry on oeis.org

0, 5, 0, 2, 149, 49, 1, 12897, 2238

Offset: 3

N. J. A. Sloane

a(n) = 0 if N does not achieve 2*floor(2n/3).

M. Gardner, The Last Recreations, Springer, 1997, p. 282.
M. Gardner, The Colossal Book of Mathematics, 2001, p. 209.

Martin Gardner, The Last Recreations, 1997
Vaclav Kotesovec, The unique solution for chessboard 9 X 9
Manfred Scheucher, C Code

Cf. A051568-A051571, A051754-A051759, A019654.

The number of solutions when N takes its maximal value is A051757.

Description corrected by and one more term from Jud McCranie, Aug 25 2001

A051754 Consider problem of placing N queens on an n X n board so that each queen attacks precisely 1 other. Sequence gives maximal number of queens.

Original entry on oeis.org

2, 2, 4, 4, 8, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 26, 28, 28, 30, 32, 32, 34, 36, 36, 38, 40, 40, 42, 44, 44, 46, 48, 48, 50, 52, 52, 54, 56, 56, 58, 60, 60, 62, 64, 64, 66, 68, 68, 70, 72, 72, 74, 76, 76, 78, 80, 80, 82, 84, 84, 86

Offset: 2

Robert Trent (trentrd(AT)hotmail.com), Aug 23 2000

2*[2n/3] is an upper bound for a(n). - Jud McCranie, Aug 12 2001
This bound is achieved for n=2, 4 and 6-65.
Conjecture: a(n) = 2*[2n/3] for n >= 6. - Alexander D. Healy, Feb 10 2024

Martin Gardner, The Last Recreations, Copernicus, NY, 1997, 274-283.

Alexander D. Healy, Examples of optimal placements for n <= 65

Cf. A051755-A051759, A051567-A051571, A019654, A063724.

a(12)-a(65) from Alexander D. Healy, Feb 10 2024

A051759 Consider the problem of placing A051756(n) queens on an n X n board so that each queen attacks precisely 3 others. Sequence gives number of solutions up to square symmetry.

Original entry on oeis.org

1, 2, 4, 31, 307, 2, 9, 755

Offset: 2

Robert Trent (trentrd(AT)hotmail.com), Aug 23 2000

A051569 (taken from the Martin Gardner references) is another version of this sequence. - N. J. A. Sloane, May 22 2014

M. Gardner, The Last Recreations, Springer, 1997, p. 282.
M. Gardner, The Colossal Book of Mathematics, 2001, p. 209.

Erich Friedman, Illustration of initial terms
Sean A. Irvine, Java program (github)

Cf. A051754-A051758, A051567-A051571, A019654.

a(8)-a(9) from Sean A. Irvine, Oct 01 2021

A051571 Consider problem of placing N queens on an n X n board so that each queen attacks precisely k others. Here k=4 and sequence gives number of different solutions when N takes its maximal value.

Original entry on oeis.org

0, 0, 0, 1, 3, 40, 615, 16573

Offset: 3

N. J. A. Sloane

M. Gardner, The Last Recreations, Springer, 1997, p. 282.

Cf. A051567-A051570, A051754-A051759, A019654.

Probably an incorrect version of A019654.

A051755 Consider problem of placing N queens on an n X n board so that each queen attacks precisely 2 others. Sequence gives maximal number of queens.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130

Offset: 2

Robert Trent (trentrd(AT)hotmail.com), Aug 23 2000

3 followed by the positive even integers starting with 4. - Wesley Ivan Hurt, Feb 09 2014

Peter Hayes, A Problem of Chess Queens, Journal of Recreational Mathematics, 24(4), 1992, 264-271.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A051754-A051759, A051567-A051571, A019654.

A051755:=n->`if`(n=2, 3, 2*n-2); seq(A051755(n), n=2..50); # Wesley Ivan Hurt, Feb 09 2014

CoefficientList[Series[(z^2 - 2*z + 3)/(z - 1)^2, {z, 0, 100}], z] (* and *) Join[{3}, Table[2*n, {n, 2, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
LinearRecurrence[{2,-1},{3,4,6},70] (* Harvey P. Dale, Aug 29 2017 *)

Vec(x^2*(x^2-2*x+3)/(x-1)^2 + O(x^100)) \\ Colin Barker, Nov 08 2014

a(2) = 3, a(n) = 2n-2 for n >= 3.
a(n) = 2*a(n-1)-a(n-2) for n>4. - Colin Barker, Nov 08 2014
G.f.: x^2*(x^2-2*x+3) / (x-1)^2. - Colin Barker, Nov 08 2014

More terms from Jud McCranie, Aug 11 2001

A051756 Consider the problem of placing N queens on an n X n board so that each queen attacks precisely 3 others. Sequence gives maximal number of queens.

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 34, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 88, 90, 92, 94, 96, 100, 102, 104, 106, 108, 112, 114, 116, 118, 120, 124, 126, 128, 130, 132, 136, 138, 140, 142, 144

Offset: 2

Robert Trent (trentrd(AT)hotmail.com), Aug 23 2000

a(n) <= 2[(6n-2)/5]. - Jud McCranie, Aug 12 2001

Conjecture: a(n) = 2[(6n-2)/5] for n >= 2; verified up to n = 100. - Alexander D. Healy, Feb 11 2024

			Examples from _R. J. Mathar_, May 01 2006: (Start)
==== n = 3
6 queens:
Q Q Q
Q - -
Q - Q
6 queens:
Q Q Q
- - -
Q Q Q
==== n = 4
8 queens:
Q Q Q Q
Q - - -
Q - - -
Q - - Q
8 queens:
Q Q Q Q
Q - - -
- - Q -
Q - - Q
8 queens:
Q Q Q Q
- - - -
- - - -
Q Q Q Q
8 queens:
Q Q - Q
- Q - -
- - Q -
Q - Q Q
==== n = 7
16 queens:
Q Q Q - Q - Q
- - - - - - Q
- - - Q - - -
Q - - - - - Q
- - - Q - - -
Q - - - - - -
Q - Q - Q Q Q
16 queens:
Q Q Q - - Q Q
- - - Q - - -
- - - - - - Q
Q - - - - - Q
Q - - - - - -
- - - Q - - -
Q Q - - Q Q Q
(End)

Martin Gardner, The Last Recreations, Copernicus, NY, 1997, 274-283.
Peter Hayes, A Problem of Chess Queens, Journal of Recreational Mathematics, Baywood, 24(4), 1992, 264-271.

Alexander D. Healy, Examples of optimal placements for n <= 61

Cf. A051754, A051755, A051757, A051758, A051759, A051567, A051568, A051569, A051570, A051571, A019654.

More terms from Jud McCranie, Aug 12 2001

a(10)-a(61) from Alexander D. Healy, Feb 11 2024

A051568 Let M(n) (A051755) be the maximal number of queens that can be placed on an n X n chessboard so that each queen attacks exactly two other queens; a(n) is the number of non-equivalent solutions. "Non-equivalent" means none of the a(n) solutions can be mapped onto any other solution by board rotations through 90, 180 or 270 degrees or mirror operations along the two diagonals or center lines.

Original entry on oeis.org

4, 2, 1, 1, 5, 2, 15, 3

Offset: 3

N. J. A. Sloane, Dec 11 1999

M. Gardner, The Last Recreations, Springer, 1997, p. 282.
M. Gardner, The Colossal Book of Mathematics, 2001, p. 209.

Ken Duisenberg, Doubly Attacking Queens, POTW 2000.

Cf. A051567-A051571, A051754-A051759, A019654.

More precise definition from R. J. Mathar, Mar 13 2006

Edited by N. J. A. Sloane, May 22 2014

A051569 Consider the problem of placing N = A051756(n) queens on an n X n board so that each queen attacks precisely 3 others; a(n) gives the number of inequivalent solutions.

Original entry on oeis.org

1, 2, 4, 31, 304, 2, 9, 755, 39302

Offset: 2

N. J. A. Sloane

M. Gardner, The Last Recreations, Springer, 1997, p. 282.
M. Gardner, The Colossal Book of Mathematics, 2001, p. 209.

Cf. A051567-A051571, A051754-A051759, A019654.

I am not certain this description is correct, nor how rigorous the results are. Probably this is an incorrect version of A051759.

A051570 Consider the problem of placing N queens on an n X n board so that each queen attacks precisely k others. Here k=3 and sequence gives number of different solutions when N takes its maximal value.

Original entry on oeis.org

2, 4, 31, 307, 2, 9, 755, 39302

Offset: 3

N. J. A. Sloane

I would also like to get the sequences that gives the maximal values of N.

M. Gardner, The Last Recreations, Springer, 1997, p. 282.

Cf. A051567-A051571, A051754-A051759, A019654.

I am not certain this description is correct, nor how rigorous the results are.

A051757 Consider problem of placing A051754(n) queens on an n X n board so that each queen attacks precisely 1 other. Sequence gives number of solutions up to square symmetry.

Original entry on oeis.org

2, 7, 5, 93, 2, 149, 49, 1, 12897, 2238

Offset: 2

Robert Trent (trentrd(AT)hotmail.com), Aug 23 2000

When n is a multiple of 3, the bound of 2[2n/3] queens is so tight that in solutions with that number of queens, all attacks must be along rows or columns, making solutions rare. - Jud McCranie

Martin Gardner, The Last Recreations, Copernicus, 1997, 274-283.

Cf. A051754-A051759, A051567-A051571, A019654.

More terms from Jud McCranie, Aug 14 2001

cp's OEIS Frontend

A051567 Consider problem of placing N queens on an n X n board so that each queen attacks precisely k others. Here k=1 and sequence gives number of inequivalent solutions when N is equal to the upper bound 2*floor(2n/3).

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A051754 Consider problem of placing N queens on an n X n board so that each queen attacks precisely 1 other. Sequence gives maximal number of queens.

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A051759 Consider the problem of placing A051756(n) queens on an n X n board so that each queen attacks precisely 3 others. Sequence gives number of solutions up to square symmetry.

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A051571 Consider problem of placing N queens on an n X n board so that each queen attacks precisely k others. Here k=4 and sequence gives number of different solutions when N takes its maximal value.

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A051755 Consider problem of placing N queens on an n X n board so that each queen attacks precisely 2 others. Sequence gives maximal number of queens.

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A051756 Consider the problem of placing N queens on an n X n board so that each queen attacks precisely 3 others. Sequence gives maximal number of queens.

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A051569 Consider the problem of placing N = A051756(n) queens on an n X n board so that each queen attacks precisely 3 others; a(n) gives the number of inequivalent solutions.

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A051570 Consider the problem of placing N queens on an n X n board so that each queen attacks precisely k others. Here k=3 and sequence gives number of different solutions when N takes its maximal value.

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