A007705 -id:A007705 - cp's OEIS frontend

A370672 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board using knight moves.

1, 0, 10, 28, 0, 88, 130, 0, 238, 304, 0, 460, 250, 0, 754, 868, 0, 280, 1258, 0, 1558, 1720, 0, 2068, 1372, 0, 2650, 880, 0, 3304, 3538, 0, 1300, 4288, 0, 4828, 5110, 0, 2464, 6004, 0, 6640, 2380, 0, 7654, 3640, 0

Offset: 0

Eduard I. Vatutin, Feb 25 2024

nonn

All solutions of this type can be found using a knight moving with some displacements dx and dy starting from some cell with coordinates (x,y): (x,y) -> (x+dx,y+dy) -> (x+2*dx,y+2*dy) -> ... -> (x,y) (all operations modulo n). For n <= 11 all solutions of n nonattacking queens on n X n a toroidal board problem are solutions of this type, for n >= 13 some solutions are not of this type (see A051906 for examples).

			For n=2*2+1=5 there are 10 solutions:
.
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
| . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |
| . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
| . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
| . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
.
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
| . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
| . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
| . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
| Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
.
so a(2)=10.

Eduard I. Vatutin, Arranging of N queens on toroidal board and generating pandiagonal Latin squares using them (in Russian).
Eduard I. Vatutin, Numerical formula between number of cyclic diagonal Latin squares and number of toroidal n-queens problem solutions getting by knight movement (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A007705, A051906, A123565, A338562.

a(n) = A123565(2*n+1) * (2*n+1).

a(n) = A338562(n) / (2n)!. - Eduard I. Vatutin, Mar 13 2024

A062166 Number of ways of placing n nonattacking torus queens on 2n+1 X 2n+1 board, similar solutions count only once.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 7, 0, 26, 46, 0, 2861, 40303, 0, 6446047

Offset: 1

Matthias Engelhardt

Two n-queens solutions p and q are considered similar iff there is a factor f, 0 < f < n, satisfying gcd (f,n) = 1, such that for all k from {0, ..., n-1} q (k * f mod n) = p (k) * f mod n or q is a rotation, a reflection or a shift of such a q. In other words, also expansions are allowed which move the queen at (k, p(k)) to (f * k mod n, f * p(k) mod n).

The sequence reduces exactly the objects of A053994 and, via that sequence, these of A007705.

M. Engelhardt, The N queens problem

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

A101453 Number of inequivalent solutions to toroidal (8n+1)-queen problem under the symmetry operator R45(x,y)=( (x-y)/sqrt(2), (x+y)/sqrt(2) ).

Original entry on oeis.org

1, 0, 4, 0, 0, 192, 1792, 0, 0, 466432, 0, 33658880, 441192448

Offset: 0

Yuh-Pyng Shieh, Yung-Luen Lan, Jieh Hsiang (arping(AT)turing.csie.ntu.edu.tw), Jan 19 2005

The R45 operator is not valid on toroidal N-queen problem if 2 is not a perfect square modulo N. For example, a(3)=0 is because 2 is not a perfect square modulo 25. see A057126. Toroidal N-queen problem has no fixed points under R45 if N is not equal to 8k+1 for some integer k.

			a(5)=6 because the number of inequivalent solutions to toroidal 41-queen problem under R45 is 192.

Jieh Hsiang, Yuh-Pyng Shieh and YaoChiang Chen, "The Cyclic Complete Mappings Counting Problems", PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002, Copenhagen, Denmark, 2002/07/27-08/01.

Yuh-Pyng Shieh, Complete Mappings

Cf. A007705, A057126.

A101454 Number of inequivalent solutions to toroidal (8n+1)-queen problem under the symmetry operator R45(x,y)=( (x-y)/sqrt(2), (x+y)/sqrt(2) ), divided by 2^n.

Original entry on oeis.org

1, 0, 1, 0, 0, 6, 28, 0, 0, 911, 0, 16435, 107713

Offset: 0

Yuh-Pyng Shieh, Yung-Luen Lan, Jieh Hsiang (arping(AT)turing.csie.ntu.edu.tw), Jan 19 2005

The R45 operator is not valid on toroidal N-queen problem if 2 is not a perfect square modulo N. For example, a(3)=0 is because 2 is not a perfect square modulo 25. See A057126. Toroidal N-queen problem has no fixed points under R45 if N is not equal to 8k+1 for some integer k.

			a(5)=6 because the number of inequivalent solutions to toroidal 41-queen problem under R45 is 192 and 192 / (2^5) = 6.

Jieh Hsiang, Yuh-Pyng Shieh and YaoChiang Chen, "The Cyclic Complete Mappings Counting Problems", PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002, Copenhagen, Denmark, 2002/07/27-08/01.

Yuh-Pyng Shieh, Complete Mappings

Cf. A007705, A057126.

A137279 Number of ways of placing ceiling(n/2) nonattacking queens on an n X n Mobius chessboard.

Original entry on oeis.org

1, 4, 0, 16, 40, 192, 560, 3328, 11772, 63840, 259336, 1550976, 7169656, 42410256, 234044160, 1366190592

Offset: 1

Brett Stevens (brett(AT)math.carleton.ca), Mar 13 2008

The chessboard is an n X n standard chessboard whose left and right edges are twisted connected.

			a(4)=16 because any queen attacks all but two other squares and every solution is counted twice by enumerating all such placements.

J. Bell and B. Stevens, Results for the n-queens problem on the Mobius board, Australasian Journal of Combinatorics, vol.42, p.21 (2008).

Cf. A000170, A007705, A002562, A053994, A061989, A061990.

A168553 a(n) = 1 if it is possible to place n sets of n queens on an n X n chessboard with no two queens of the same set attacking each other.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1

Offset: 1

Howard A. Landman, Nov 29 2009

Obviously this sequence must be a subset of the positive integers for which a single n-queens solution exists, which is all integers except 2 and 3. It is a proper subset, because e.g. there is a single-set solution for 4 X 4 or 8 X 8 but no n-set solution. George Polya showed that a doubly periodic solution for the n-queens problem exists if and only if n = 1 or 5 (mod 6). For years, due to this result, it was assumed that this defined the sequence, which would then have been an infinite repetition of 1,0,0,0,1,0. However, Patrick Hamlyn and Guenter Stertenbrink tried brute force on 12 X 12 and found 178 12-set solutions built from non-doubly-periodic sets. Thus the Polya condition only provides a lower bound, and the exact continuation of this sequence is an open research question.

			For n=1 a single queen is a solution. For n=2 or 3 there is no single-set solution. For n=4, there are 2 single-set solutions, and they can both fit on the board together, but these are insufficient to make a 4-set solution. For n=5, there is a doubly-periodic single-set solution by Polya's construction, which can be tiled to make a 5-set solution

G. Polya, Uber die 'doppelt-periodischen' Losungen des n-Damen-Problems. In Mathematische Unterhaltungen und Spiele, W. Ahrens, Ed., Teubner, Leipzig, 1918, pp. 364-374.

Wolfram Research, The 12x12 Queens Problem

A000170 gives the number of single-set solutions. A007705 gives the number of Polya-style doubly-periodic single-set solutions (with even n omitted since they are all 0).

cp's OEIS Frontend

A370672 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board using knight moves.

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A062166 Number of ways of placing n nonattacking torus queens on 2n+1 X 2n+1 board, similar solutions count only once.

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A101453 Number of inequivalent solutions to toroidal (8n+1)-queen problem under the symmetry operator R45(x,y)=( (x-y)/sqrt(2), (x+y)/sqrt(2) ).

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A101454 Number of inequivalent solutions to toroidal (8n+1)-queen problem under the symmetry operator R45(x,y)=( (x-y)/sqrt(2), (x+y)/sqrt(2) ), divided by 2^n.

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A137279 Number of ways of placing ceiling(n/2) nonattacking queens on an n X n Mobius chessboard.

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A168553 a(n) = 1 if it is possible to place n sets of n queens on an n X n chessboard with no two queens of the same set attacking each other.

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