A051757 -id:A051757 - 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

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

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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