A051567 -id:A051567 - cp's OEIS frontend

A051566 Erroneous version of A051567.

Original entry on oeis.org

0, 5, 0, 2, 138, 47, 1, 12490

Offset: 3

dead

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

A019654 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, 1, 1, 1, 3, 40, 655, 16573

Offset: 3

N. J. A. Sloane, Oct 03 2002

I would also like to get the sequences that give the maximal value of N.
Maximal values given by A063724. - Sean A. Irvine, Mar 31 2019
Values for a(9) onwards currently unverified. - Sean A. Irvine, Apr 05 2019

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

Sean A. Irvine, Solutions for n<=8
Eric Weisstein's World of Mathematics, Queens Problem [From Claudio (lordofchaos000(AT)hotmail.com), May 31 2010]

Cf. A051567-A051571, A051754-A051759, A063724.

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

a(4) and a(5) changed to 1 and irrelevant comment removed by Sean A. Irvine, Apr 02 2019

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.

cp's OEIS Frontend

A051566 Erroneous version of A051567.

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A019654 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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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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