A051906 -id:A051906 - cp's OEIS frontend

A000170 Number of ways of placing n nonattacking queens on an n X n board.

1, 1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184, 14772512, 95815104, 666090624, 4968057848, 39029188884, 314666222712, 2691008701644, 24233937684440, 227514171973736, 2207893435808352, 22317699616364044, 234907967154122528

Offset: 0

N. J. A. Sloane

For n > 3, a(n) is the number of maximum independent vertex sets in the n X n queen graph. - Eric W. Weisstein, Jun 20 2017
Number of nodes on level n of the backtrack tree for the n queens problem (a(n) = A319284(n, n)). - Peter Luschny, Sep 18 2018
Number of permutations of [1...n] such that |p(j)-p(i)| != j-i for iXiangyu Chen, Dec 24 2020
M. Simkin shows that the number of ways to place n mutually nonattacking queens on an n X n chessboard is ((1 +/- o(1))*n*exp(-c))^n, where c = 1.942 +/- 0.003. These are approximately (0.143*n)^n configurations. - Peter Luschny, Oct 07 2021

			a(2) = a(3) = 0, since on 2 X 2 and 3 X 3 chessboards there are no solutions.
.
a(4) = 2:
  +---------+ +---------+
  | . . Q . | | . Q . . |
  | Q . . . | | . . . Q |
  | . . . Q | | Q . . . |
  | . Q . . | | . . Q . |
  +---------+ +---------+
a(5) = 10:
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
  | . . . 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 |
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
a(6) = 4:
  +-------------+ +-------------+ +-------------+ +-------------+
  | . . . . Q . | | . . . Q . . | | . . Q . . . | | . Q . . . . |
  | . . Q . . . | | Q . . . . . | | . . . . . Q | | . . . Q . . |
  | Q . . . . . | | . . . . Q . | | . Q . . . . | | . . . . . Q |
  | . . . . . Q | | . Q . . . . | | . . . . Q . | | Q . . . . . |
  | . . . Q . . | | . . . . . Q | | Q . . . . . | | . . Q . . . |
  | . Q . . . . | | . . Q . . . | | . . . Q . . | | . . . . Q . |
  +-------------+ +-------------+ +-------------+ +-------------+
- _Hugo Pfoertner_, Mar 17 2019

M. Gardner, The Unexpected Hanging, pp. 190-2, Simon & Shuster NY 1969
Jieh Hsiang, Yuh-Pyng Shieh and Yao-Chiang Chen, The cyclic complete mappings counting problems, in Problems and Problem Sets for ATP, volume 02-10 of DIKU technical reports, G. Sutcliffe, J. Pelletier and C. Suttner, eds., 2002.
D. E. Knuth, The Art of Computer Programming, Volume 4, Pre-fascicle 5B, Introduction to Backtracking, 7.2.2. Backtrack programming. 2018.
M. Kraitchik, The Problem of The Queens, Mathematical Recreations, 2nd ed., New York, Dover, 1953, pp. 247-256.
Massimo Nocentini, "An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation", PhD Thesis, University of Florence, 2019. See Ex. 67.
W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York, Dover, 1987, pp. 166-172 (The Eight Queens Problem).
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
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).
R. J. Walker, An enumerative technique for a class of combinatorial problems, pp. 91-94 of Proc. Sympos. Applied Math., vol. 10, Amer. Math. Soc., 1960.
M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.

Jordan Bell and Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309, Issue 1, Jan 06 2009, Pages 1-31.
D. Bill, Durango Bill's The N-Queens Problem
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656.
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656. [Annotated scanned copy]
Candida Bowtell and Peter Keevash, The n-queens problem, arXiv:2109.08083 [math.CO] 2021.
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
V. Chvatal, All solutions to the problem of eight queens
V. Chvatal, All solutions to the problem of eight queens [Cached copy, pdf format, with permission]
Gheorghe Coserea, Solutions for n=8.
Gheorghe Coserea, Solutions for n=9.
Gheorghe Coserea, MiniZinc model for generating solutions.
Matteo Fischetti and Domenico Salvagnin, Chasing First Queens by Integer Programming, 2018.
Matteo Fischetti and Domenico Salvagnin, Finding First and Most-Beautiful Queens by Integer Programming, arXiv:1907.08246 [cs.DS], 2019.
J. Freeman, A neural network solution to the n-queens problem, The Mathematica J., 3 (No. 3, 1993), 52-56.
Ian P. Gent, Christopher Jefferson and Peter Nightingale, Complexity of n-Queens Completion, Journal of Artificial Intelligence Research 59 (2017), see p 816.
Eric Grigoryan, Investigation of the Regularities in the Formation of Solutions n-Queens Problem, Modeling of Artificial Intelligence, 2018, 5(1), 3-21.
E. Grigoryan, Linear algorithm for solution n-Queens Completion problem, arXiv:1912.05935 [cs.AI], 2019.
James Grime and Brady Haran, The 8 Queen Problem, Numberphile video (2015).
Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.
Kenji Kise, Takahiro Katagiri, Hiroki Honda and Toshitsugu Yuba, Solving the 24-queens Problem using MPI on a PC Cluster, Technical Report UEC-IS-2004-6, Graduate School of Information Systems, The University of Electro-Communications (2004).
D. E. Knuth, Donald Knuth's 24th Annual Christmas Lecture: Dancing Links, Stanfordonline, Video published on YouTube, Dec 12, 2018.
W. Kosters, n-Queens bibliography
Vaclav Kotesovec, Non-attacking chess pieces, Sixth edition, 795 pages, Feb 02 2013 (minor update Mar 29 2016).
Zur Luria, New bounds on the number of n-queens configurations, arXiv:1705.05225 [math.CO], 2017.
Zur Luria, Michael Simkin, A lower bound for the n-queens problem, arXiv:2105.11431 [math.CO], 2021.
E. Masehian, H. Akbaripour and N. Mohabbati-Kalejahi, Solving the n Queens Problem using a Tuned Hybrid Imperialist Competitive Algorithm, 2013.
E. Masehian, H. Akbaripour and N. Mohabbati-Kalejahi, Landscape analysis and efficient metaheuristics for solving the n-queens problem, Computational Optimization and Applications, 2013; DOI 10.1007/s10589-013-9578-z.
Nasrin Mohabbati-Kalejahi, Hossein Akbaripour and Ellips Masehian, Basic and Hybrid Imperialist Competitive Algorithms for Solving the Non-attacking and Non-dominating n -Queens Problems, Studies in Computational Intelligence Volume 577, 2015, pp 79-96. DOI 10.1007/978-3-319-11271-8_6.
Ralph Morrison and Noah Speeter, The Gonality of Queen's Graphs, arXiv:2312.04686 [math.CO], 2023.
Parth Nobel, Akshay Agrawal and Stephen Boyd, Computing tighter bounds on the n-queens constant via Newton’s method, arXiv:2112.03336 [math.CO], 2021.
J. Pope and D. Sonnier, A linear solution to the n-Queens problem using vector spaces, Journal of Computing Sciences in Colleges, Volume 29 Issue 5, May 2014 Pages 77-83.
T. B. Preußer and M. R. Engelhardt, Putting Queens in Carry Chains, No. 27, Journal of Signal Processing Systems, Volume 88, Issue 2, August 2017. (The title refers to the fact that the article discusses the case n = 27.)
Thomas B. Preußer, Bernd Nägel and Rainer G. Spallek, Putting Queens in Carry Chains, Slides, HIPEAC WRC'09.
E. M. Reingold, Letter to N. J. A. Sloane, Dec 27 1973
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Werner Florian Samayoa, Maria Liz Crespo, Sergio Carrato, Agustin Silva, and Andres Cicuttin, HyperFPGA: An Experimental Testbed for Heterogeneous Supercomputing, 2023.
Michael Simkin, The number of n-queens configurations, arXiv:2107.13460 [math.CO], 2021.
Wenxi Wang, Muhammad Usman, Alyas Almaawi, Kaiyuan Wang, Kuldeep S. Meel and Sarfraz Khurshid, A Study of Symmetry Breaking Predicates and Model Counting, National University of Singapore (2020).
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
Eric Weisstein's World of Mathematics, Queen Graph
Eric Weisstein's World of Mathematics, Queens Problem
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Wikipedia, Eight Queens Puzzle
Cheng Zhang and Jianpeng Ma, Counting Solutions for the N-queens and Latin Square Problems by Efficient Monte Carlo Simulations, arXiv:0808.4003 [cond-mat.stat-mech], 2008.

Cf. A002562, A065256.
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A099152, A006717, A051906, A319284 (backtrack trees).
Cf. A260318, A260319, A260320.
Main diagonal of A348129.

Strong conjecture: there is a constant c around 2.54 such that a(n) is asymptotic to n!/c^n; weak conjecture: lim_{n -> infinity} (1/n) * log(n!/a(n)) = constant = 0.90.... - Benoit Cloitre, Nov 10 2002
Lim_{n->infinity} a(n)^(1/n)/n = exp(-A359441) = 0.1431301... [Simkin 2021]. - Vaclav Kotesovec, Jan 01 2023
a(n) = 8 * A260320(n) + 4 * A260319(n) + 2 * A260318(n) for n >= 2 (see Kraitchik reference). - Jason Bard, Aug 12 2025

Terms for n=21-23 computed by Sylvain PION (Sylvain.Pion(AT)sophia.inria.fr) and Joel-Yann FOURRE (Joel-Yann.Fourre(AT)ens.fr).
a(24) from Kenji KISE (kis(AT)is.uec.ac.jp), Sep 01 2004
a(25) from Objectweb ProActive INRIA Team (proactive(AT)objectweb.org), Jun 11 2005 [Communicated by Alexandre Di Costanzo (Alexandre.Di_Costanzo(AT)sophia.inria.fr)]. This calculation took about 53 years of CPU time.
a(25) has been confirmed by the NTU 25Queen Project at National Taiwan University and Ming Chuan University, led by Yuh-Pyng (Arping) Shieh, Jul 26 2005. This computation took 26613 days CPU time.
The NQueens-at-Home web site gives a different value for a(24), 226732487925864. Thanks to Goran Fagerstrom for pointing this out. I do not know which value is correct. I have therefore created a new entry, A140393, which gives the NQueens-at-home version of the sequence. - N. J. A. Sloane, Jun 18 2008
It now appears that this sequence (A000170) is correct and A140393 is wrong. - N. J. A. Sloane, Nov 08 2008
Added a(26) as calculated by Queens(AT)TUD [http://queens.inf.tu-dresden.de/]. - Thomas B. Preußer, Jul 11 2009
Added a(27) as calculated by the Q27 Project [https://github.com/preusser/q27]. - Thomas B. Preußer, Sep 23 2016
a(0) = 1 prepended by Joerg Arndt, Sep 16 2018

A007705 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board.

Original entry on oeis.org

1, 0, 10, 28, 0, 88, 4524, 0, 140692, 820496, 0, 128850048, 1957725000, 0, 605917055356, 13404947681712, 0

Offset: 0

N. J. A. Sloane

Polya proved (see Ahrens) that the number of solutions to this problem for an m X m board is > 0 iff m is coprime to 6. - Jonathan Vos Post, Feb 20 2005

			From _Eduard I. Vatutin_, Jan 22 2024: (Start)
N=5=2*2+1 (all 10 solutions are shown below):
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| 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 . . . |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
N=7=2*3+1:
+---------------+
| Q . . . . . . |
| . . . Q . . . |
| . . . . . . Q |
| . . Q . . . . |
| . . . . . Q . |
| . Q . . . . . |
| . . . . Q . . |
+---------------+
N=11=5*2+1:
+-----------------------+
| Q . . . . . . . . . . |
| . . Q . . . . . . . . |
| . . . . Q . . . . . . |
| . . . . . . Q . . . . |
| . . . . . . . . Q . . |
| . . . . . . . . . . Q |
| . Q . . . . . . . . . |
| . . . Q . . . . . . . |
| . . . . . Q . . . . . |
| . . . . . . . Q . . . |
| . . . . . . . . . Q . |
+-----------------------+
N=13=6*2+1 (first example can be found using a knight moving from cell (1,1) with dx=1 and dy=2, second example can't be obtained in the same way):
+---------------------------+ +---------------------------+
| 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 . . . . |
+---------------------------+ +---------------------------+
(End)

W. Ahrens, Mathematische Unterhaltungen und Spiele, Vol. 1, B. G. Teubner, Leipzig, 1921, pp. 363-374.
R. K. Guy, Unsolved problems in Number Theory, 3rd Edn., Springer, 1994, p. 202 [with extensive bibliography]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesly, 1991, Chapter 6.

M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 62-63.
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Eric Weisstein's World of Mathematics, Queens Problem.
Eduard I. Vatutin, Numerical formula between number of horizontally or vertically semicyclic diagonal Latin squares and number of toroidal n-queens problem solutions (in Russian).

Cf. A051906, A071607, A342990.

a(n) = A071607(n) * (2*n+1). - Eduard I. Vatutin, Jan 22 2024, corrected Mar 14 2024

a(n) = A342990(n) / (2n)!. - Eduard I. Vatutin, Apr 09 2024

Two more terms from Matthias Engelhardt, Dec 17 1999 and Jan 11 2001

13404947681712 from Matthias Engelhardt, May 01 2005

A123565 a(n) is the number of positive integers k which are <= n and where k, k-1 and k+1 are each coprime to n.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 4, 0, 0, 0, 8, 0, 10, 0, 0, 0, 14, 0, 16, 0, 0, 0, 20, 0, 10, 0, 0, 0, 26, 0, 28, 0, 0, 0, 8, 0, 34, 0, 0, 0, 38, 0, 40, 0, 0, 0, 44, 0, 28, 0, 0, 0, 50, 0, 16, 0, 0, 0, 56, 0, 58, 0, 0, 0, 20, 0, 64, 0, 0, 0, 68, 0, 70, 0, 0, 0, 32, 0, 76, 0, 0, 0, 80, 0, 28, 0, 0, 0, 86, 0, 40, 0, 0

Offset: 1

Leroy Quet, Nov 12 2006

a(p) = p-3 for any odd prime p. a(2n) = a(3n) = 0.
a(n) > 0 if and only if n is coprime to 6. - Chai Wah Wu, Aug 26 2016
Multiplicative by the Chinese remainder theorem. - Andrew Howroyd, Aug 07 2018
From Eduard I. Vatutin, Nov 03 2020: (Start)
a(n) is the number of cyclic diagonal Latin squares of order n with the first row in order. Every cyclic diagonal Latin square is a cyclic Latin square, so a(n) <= A000010(n). Every cyclic diagonal Latin square is pandiagonal, but the converse is not true. For example, for order n=13 there is a square
   7  1  0  3  6  5 12  2  8  9 10 11  4
   2  3  4 10  0  7  6  9 12 11  5  8  1
   4 11  1  7  8  9 10  3  6  0 12  2  5
   6  5  8 11 10  4  7  0  1  2  3  9 12
   8  9  2  5 12 11  1  4  3 10  0  6  7
   3  6 12  0  1  2  8 11  5  4  7 10  9
  10  0  3  2  9 12  5  6  7  8  1  4 11
   1  7 10  4  3  6  9  8  2  5 11 12  0
  11  4  5  6  7  0  3 10  9 12  2  1  8
   5  8  7  1  4 10 11 12  0  6  9  3  2
  12  2  9  8 11  1  0  7 10  3  4  5  6
   9 10 11 12  5  8  2  1  4  7  6  0  3
   0 12  6  9  2  3  4  5 11  1  8  7 10
that is pandiagonal but not cyclic (Dabbaghian and Wu). (End)
Schemmel's totient function of order 3 (Schemmel, 1869; Sándor and Crstici, 2004). - Amiram Eldar, Nov 22 2020
a(p) is a lower bound for cardinality of clique of MODLS for all odd prime orders p: a(p) <= A328873(p). - Eduard I. Vatutin, Apr 02 2021
Also number of solutions for n-queens problem on toroidal chessboard (see A051906, A007705 or A370672), given by knight with (dx,dy) movement parameters starting from top left corner (more generally: from one cell fixed for all solutions). - Eduard I. Vatutin, Mar 13 2024

			The positive integers which are both coprime to 25 and are <= 25 are 1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,21,22,23,24. Of these integers there are 10 integers k where (k-1) and (k+1) are also coprime to 25. These integers k are 2,3,7,8,12,13,17,18,22,23. So a(25) = 10.
Example of a cyclic diagonal Latin square of order 5:
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
Example of a cyclic diagonal Latin square of order 7:
  0 1 2 3 4 5 6
  2 3 4 5 6 0 1
  4 5 6 0 1 2 3
  6 0 1 2 3 4 5
  1 2 3 4 5 6 0
  3 4 5 6 0 1 2
  5 6 0 1 2 3 4
From _Eduard I. Vatutin_, Mar 13 2024: (Start)
Example of a(5)=2 solutions for n-queens problem on toroidal chessboard, given by knight with (+1,+2) and (+1,+3) movement parameters starting from top left corner:
.
+-----------+ +-----------+
| Q . . . . | | Q . . . . |
| . . Q . . | | . . . Q . |
| . . . . Q | | . Q . . . |
| . Q . . . | | . . . . Q |
| . . . Q . | | . . Q . . |
+-----------+ +-----------+
.
Example of a(7)=4 solutions for n-queens problem on toroidal chessboard, given by knight with (+1,+2), (+1,+3), (+1,+4), (+1,+5) movement parameters starting from top left corner:
.
+---------------+ +---------------+ +---------------+ +---------------+
| 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 . . . . |
+---------------+ +---------------+ +---------------+ +---------------+
(End)

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 276.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vahid Dabbaghian and Tiankuang Wu, Constructing non-cyclic pandiagonal Latin squares of prime orders, Journal of Discrete Algorithms, Vol. 30 (2015), pp. 70-77.
Colin Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq., Vol. 18 (2015), Article # 15.2.1.
A. Hedayat, A complete solution to the existence and nonexistence of Knut Vik designs and orthogonal Knut Vik designs, J. Comb. Theory, Ser. A 22(3) (1977) 331-337.
Victor Schemmel, Ueber relative Primzahlen, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 191-192.
Eduard I. Vatutin, Enumerating cyclic Latin squares and Euler totient function calculating using them, High-performance computing systems and technologies, 2020, Vol. 4, No. 2, pp. 40-48. (in Russian)
Eduard I. Vatutin, Arranging of N queens on toroidal board and generating pandiagonal Latin squares using them (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A000010, A007705, A051906, A058026, A206256, A241663, A328873, A370672.

f:= proc(n) local V,R;
  V:= map(igcd,[$1..n],n);
  R:= V[1..n-2] + V[2..n-1] + V[3..n];
  numboccur(3,R);
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Mar 15 2024

f[n_] := Length[Select[Range[n],GCD[ #, n] == 1 && GCD[ # - 1, n] == 1 && GCD[ # + 1, n] == 1 &]];Table[f[n], {n, 100}] (* Ray Chandler, Nov 19 2006 *)
Join[{1},Table[Count[Boole[Partition[CoprimeQ[Range[n],n],3,1]],{1,1,1}],{n,2,100}]] (* Harvey P. Dale, Apr 09 2017 *)
f[2, e_] := 0; f[p_, e_] := (p - 3)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 22 2020 *)

a(n)=if(gcd(n,6)>1, return(0)); sum(k=1,n,gcd(k^3-k,n)==1) \\ Charles R Greathouse IV, Aug 26 2016

Multiplicative with a(2^e) = 0 and a(p^e) = (p-3)*p^(e-1) for odd primes p. - Amiram Eldar, Nov 22 2020
a(2*n+1) = A338562(n) / (2*n+1)!. - Eduard I. Vatutin, Apr 02 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (1 - 3/p^2) = 0.125486... (A206256). - Amiram Eldar, Nov 18 2022
a(n) = A370672((n-1)/2) / n. - Eduard I. Vatutin, Mar 13 2024

Extended by Ray Chandler, Nov 19 2006

A071607 Number of strong complete mappings of the cyclic group Z_{2n+1}.

Original entry on oeis.org

1, 0, 2, 4, 0, 8, 348, 0, 8276, 43184, 0, 5602176, 78309000, 0, 20893691564, 432417667152, 0

Offset: 0

J. Hsiang, D. F. Hsu and Y. P. Shieh (arping(AT)turing.csie.ntu.edu.tw), Jun 03 2002

A strong complete mapping of a cyclic group (Z_n,+) is a permutation f(x) of Z_n such that f(0)=0 and that f(x)-x and f(x)+x are both permutations.
a(n) is the number of solutions of the toroidal n-queen problem (A007705) with 2n+1 queens and one queen in the top-left corner.
Also a(n) is the number of horizontally or vertically semicyclic diagonal Latin squares of order 2n+1 with the first row in ascending order. Horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row  r(0) by some value d(i) (see example). Vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column  c(0) by some value d(i). Cyclic diagonal Latin squares (see A123565) fall under the definition of vertically and horizontally semicyclic diagonal Latin squares simultaneously, in this type of squares each row r(i) is obtained from the previous one r(i-1) using cyclic shift by some value d. Definition from A343867 includes this type of squares but not only it. - Eduard I. Vatutin, Jan 25 2022

			f(x)=2x in (Z_7,+) is a strong complete mapping of Z_7 since f(0)=0 and both f(x)-x (=x) and f(x)+x (=3x) are permutations of Z_7.
From _Eduard I. Vatutin_, Jan 25 2022: (Start)
Example of cyclic diagonal Latin square of order 13:
.
   0  1  2  3  4  5  6  7  8  9 10 11 12
   2  3  4  5  6  7  8  9 10 11 12  0  1  (d=2)
   4  5  6  7  8  9 10 11 12  0  1  2  3  (d=4)
   6  7  8  9 10 11 12  0  1  2  3  4  5  (d=6)
   8  9 10 11 12  0  1  2  3  4  5  6  7  (d=8)
  10 11 12  0  1  2  3  4  5  6  7  8  9  (d=10)
  12  0  1  2  3  4  5  6  7  8  9 10 11  (d=12)
   1  2  3  4  5  6  7  8  9 10 11 12  0  (d=1)
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=3)
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=5)
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=7)
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=9)
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=11)
.
Example of horizontally semicyclic diagonal Latin square of order 13:
.
   0  1  2  3  4  5  6  7  8  9 10 11 12
   2  3  4  5  6  7  8  9 10 11 12  0  1  (d=2)
   4  5  6  7  8  9 10 11 12  0  1  2  3  (d=4)
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=9)
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=7)
  12  0  1  2  3  4  5  6  7  8  9 10 11  (d=12)
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=3)
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=11)
   6  7  8  9 10 11 12  0  1  2  3  4  5  (d=6)
   1  2  3  4  5  6  7  8  9 10 11 12  0  (d=1)
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=5)
  10 11 12  0  1  2  3  4  5  6  7  8  9  (d=10)
   8  9 10 11 12  0  1  2  3  4  5  6  7  (d=8)
(End)
From _Eduard I. Vatutin_, Apr 09 2024: (Start)
Example of N-queens problem on toroidal board, N=2*2+1=5, a(2)=2, given by knight with (+1,+2) and (+1,+3) movement parameters starting from top left corner:
.
+-----------+ +-----------+
| Q . . . . | | Q . . . . |
| . . Q . . | | . . . Q . |
| . . . . Q | | . Q . . . |
| . Q . . . | | . . . . Q |
| . . . Q . | | . . Q . . |
+-----------+ +-----------+
.
Example of N-queens problem on toroidal board, N=2*3+1=7, a(3)=4, given by knight with (+1,+2), (+1,+3), (+1,+4), (+1,+5) movement parameters starting from top left corner:
.
+---------------+ +---------------+ +---------------+ +---------------+
| 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 . . . . |
+---------------+ +---------------+ +---------------+ +---------------+
(End)

Anthony B. Evans,"Orthomorphism Graphs of Groups", vol. 1535 of Lecture Notes in Mathematics, Springer-Verlag, 1991.
Y. P. Shieh, J. Hsiang and D. F. Hsu, "On the enumeration of Abelian k-complete mappings", vol. 144 of Congressus Numerantium, 2000, pp. 67-88.

Jieh Hsiang, Yuh Pyng Shieh, and Yao Chiang Chen, Cyclic complete mappings counting problems, in PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002.
Jieh Hsiang, YuhPyng Shieh, and YaoChiang Chen, Cyclic Complete Mappings Counting Problems, National Taiwan University 2014/8/21.
D. Novakovic, Computation of the number of complete mappings for permutations, Cybernetics & System Analysis, No. 2, v. 36 (2000), pp. 244-247.
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), pp. 629-639.
Eduard I. Vatutin, About the horizontally and vertically semicyclic diagonal Latin squares enumeration (in Russian).
Eduard I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A007705, A051906, A123565, A338620, A342990, A343867.

a(n) = A007705(n) / (2*n+1).
a(n) = A342990(n) / (2*n+1)!. - Eduard I. Vatutin, Mar 10 2022
a(n) = A051906(2*n+1) / (2*n+1). - Eduard I. Vatutin, Apr 09 2024

a(15)-a(16) added using A007705 by Andrew Howroyd, May 07 2021

A085801 Maximum number of nonattacking queens on an n X n toroidal board.

Original entry on oeis.org

1, 1, 1, 2, 5, 4, 7, 6, 7, 9, 11, 10, 13, 13, 13, 14, 17, 16, 19, 18, 19, 21, 23, 22, 25, 25, 25, 26, 29, 28, 31, 30, 31, 33, 35, 34, 37, 37, 37, 38, 41, 40, 43, 42, 43, 45, 47, 46, 49, 49, 49, 50, 53, 52, 55, 54, 55, 57, 59, 58, 61, 61, 61, 62, 65, 64, 67, 66

Offset: 1

Konrad Schlude, Jul 24 2003

Independence number of the queens' graph on toroidal n X n board. - Andrey Zabolotskiy, Dec 11 2016

			Four non-attacking queens can be placed on a 6 X 6 toroidal board:
......
..Q...
....Q.
.Q....
...Q..
......
But five queens cannot. Hence a(6) = 4.

G. Polya: Über die 'Doppelt-Periodischen' Loesungen des n-Damen-Problems, in: W. Ahrens: Mathematische Unterhaltungen und Spiele, Teubner, Leipzig, 1918, 364-374. Reprinted in: G. Polya: Collected Works, Vol. V, 237-247.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Grant Cairns, Queens on Non-square Tori, El. J. Combinatorics, N6, 2001
Eldar Fischer, Tomer Kotek, and Johann A. Makowsky, Application of Logic to Combinatorial Sequences and Their Recurrence Relations
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 751.
P. Monsky, Problem E3162, Amer. Math. Monthly 96 (1989), 258-259.
Konrad Schlude and Ernst Specker, Zum Problem der Damen auf dem Torus, Technical Report 412, Computer Science Department ETH Zurich, 2003.

Cf. A051906, A007705, A279402, A279404, A279405, A172517, A172518, A172519, A173775, A178722.

(* Explicit formula, based on an article by Monsky: *)
Table[n-1/6*(2*Cos[Pi*n/2]-3*Cos[Pi*n/3]+5*Cos[2*Pi*n/3]-Cos[Pi*n/6]-Cos[5*Pi*n/6]+3*Cos[Pi*n]+7),{n,1,100}] (* Vaclav Kotesovec, Dec 13 2010 *)

a(n)=n-1/6*(2*cos(Pi*n/2)-3*cos(Pi*n/3)+5*cos(2*Pi*n/3)-cos(Pi*n/6)-cos(5*Pi*n/6)+3*cos(Pi*n)+7);
vector(60,n,round(a(n))) \\ Joerg Arndt, Dec 13 2010

G.f.: (2*x^12 - x^11 + 2*x^10 + 2*x^9 + x^8 - x^7 + 3*x^6 - x^5 + 3*x^4 + x^3 + 1)/(x^13 - x^12 - x + 1) = (2*x^12 - x^11 + 2*x^10 + 2*x^9 + x^8 - x^7 + 3*x^6 - x^5 + 3*x^4 + x^3 + 1)/((x - 1)^2*(x + 1)*(x^2 + 1)*(x^2 - x + 1)*(x^2 + x + 1)*(x^4 - x^2 + 1)). - Joerg Arndt, Dec 13 2010
From Andrey Zabolotskiy, Dec 11 2016: (Start)
a(n) = n if n = 1, 5, 7, 11 (mod 12);
a(n) = n-1 if n = 2, 10 (mod 12);
a(n) = n-2 otherwise.
(End)

A053994 Nonattacking queens on a 2n+1 X 2n+1 toroidal board, solutions which differ only by rotation, reflection or torus shift count only once.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 11, 0, 97, 354, 0, 31381, 395551, 0, 90120677

Offset: 0

Matthias Engelhardt, Apr 05 2000

A. P. Street and R. Day, Sequential binary arrays II: Further results on the square grid, pp. 392-418 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982 (for getting equivalence classes).

I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639 (for finding the solutions).

A007705, A006841, A051906.

More terms from Matthias Engelhardt, Jan 11 2001

A190393 Number of ways to place n nonattacking nightriders on an n X n toroidal board.

Original entry on oeis.org

1, 2, 6, 24, 120, 144, 28, 1408, 2025, 86400, 1782, 1092096, 4186, 31360, 241920000, 23953408, 140692, 114108912, 1092690

Offset: 1

Vaclav Kotesovec, May 10 2011

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Vaclav Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech).

Cf. A172141, A173429, A051906.

Terms a(16)-a(17) from Vaclav Kotesovec, May 14 2011

Terms a(18)-a(19) from Vaclav Kotesovec, May 28 2011

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

Original entry on oeis.org

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

cp's OEIS Frontend

A000170 Number of ways of placing n nonattacking queens on an n X n board.

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A007705 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board.

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A123565 a(n) is the number of positive integers k which are <= n and where k, k-1 and k+1 are each coprime to n.

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A071607 Number of strong complete mappings of the cyclic group Z_{2n+1}.

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A085801 Maximum number of nonattacking queens on an n X n toroidal board.

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A053994 Nonattacking queens on a 2n+1 X 2n+1 toroidal board, solutions which differ only by rotation, reflection or torus shift count only once.

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A190393 Number of ways to place n nonattacking nightriders on an n X n toroidal board.

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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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