A036464 -id:A036464 - 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

A024023 a(n) = 3^n - 1.

Original entry on oeis.org

0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442, 2541865828328, 7625597484986, 22876792454960

Offset: 0

N. J. A. Sloane

Number of different directions along lines and hyper-diagonals in an n-dimensional cubic lattice for the attacking queens problem (A036464 in n=2, A068940 in n=3 and A068941 in n=4). The n-dimensional direction vectors have the a(n)+1 Cartesian coordinates (i,j,k,l,...) where i,j,k,l,... = -1, 0, or +1, excluding the zero-vector i=j=k=l=...=0. The corresponding hyper-line count is A003462. - R. J. Mathar, May 01 2006
Total number of sequences of length m=1,...,n with nonzero integer elements satisfying the condition Sum_{k=1..m} |n_k| <= n. See the K. A. Meissner link p. 6 (with a typo: it should be 3^([2a]-1)-1). - Wolfdieter Lang, Jan 21 2008
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x and y are disjoint and either 0) x is a proper subset of y or y is a proper subset of x, or 1) x is not a subset of y and y is not a subset of x. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
Number of neighbors in Moore's neighborhood in n dimensions. - Dmitry Zaitsev, Nov 30 2015
Number of terms in conjunctive normal form of Boolean expression with n variables. E.g., a(2) = 8: [~x, ~y, x, y, ~x|~y, ~x|y, x|~y, x|y]. - Yuchun Ji, May 12 2023
Number of rays of the Coxeter arrangement of type B_n. Equivalently, number of facets of the n-dimensional type B permutahedron. - Jose Bastidas, Sep 12 2023

			From _Zerinvary Lajos_, Jan 14 2007: (Start)
Ternary......decimal:
0...............0
2...............2
22..............8
222............26
2222...........80
22222.........242
222222........728
2222222......2186
22222222.....6560
222222222...19682
2222222222..59048
etc...........etc.
(End)
Sequence combinatorics: n=3: With length m=1: [1],[2],[3] each with 2 signs, with m=2: [1,1], [1,2], [2,1], each 2^2 = 4 times from choosing signs; m=3: [1,1,1] coming in 2^3 signed versions: 3*2 + 3*4 + 1*8 = 26 = a(3). The order is important, hence the M_0 multinomials A048996 enter as factors.
A027902 gives the 384 divisors of a(24). - _Reinhard Zumkeller_, Mar 11 2010

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Omran Ahmadi and Robert Granger, An efficient deterministic test for Kloosterman sum zeros, Mathematics of Computation, Vol. 83, No. 285 (2014), pp. 347-363; arXiv preprint, arXiv:1104.3882 [math.NT], 2011-2012. See 1st column of Table 2, p. 9.
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Michael Baake, Franz Gähler, and Uwe Grimm, Examples of Substitution Systems and Their Factors, Journal of Integer Sequences, Vol. 16 (2013), #13.2.14.
R. Samuel Buss, Herbrand's Theorem, University of California, Logic and Computational Complexity pp. 195-209, Lecture Notes in Computer Science, vol 960. Springer.
Jan Draisma, Tyrrell B. McAllister and Benjamin Nill, Lattice width directions and Minkowski's 3^d-theorem, SIAM J. Discrete Math., Vol. 26, No. 3 (2012), pp. 1104-1107; arXiv preprint, arXiv:0901.1375 [math.CO], Jan 10 2009.
Alessandro Farinelli, Herbrand Universe and Herbrand Base.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Krzysztof A. Meissner, Black hole entropy in Loop Quantum Gravity, Classical and Quantum Gravity, Vol. 21, No. 22 (2004), pp. 5245--5251; arXiv preprint, arXiv:gr-qc/0407052, 2004.
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case three-in-a row, sequence b(n).
Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Wikipedia, Herbrand Structure.
Damiano Zanardini, Computational Logic, Slides, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid, 2009-2010.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. triangle A013609.
Cf. A003462, A007051, A034472, A214369.
Cf. second column of A145901.

a024023 = subtract 1 . a000244  -- Reinhard Zumkeller, Jun 30 2013

[3^n-1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

3^Range[0,30]-1 (* Paolo Xausa, Jul 15 2023 *)

a(n)=3^n-1 \\ Charles R Greathouse IV, Sep 24 2015

vector(50, n, sum(k=0, n, 2^k*binomial(n-1, k))-1) \\ Altug Alkan, Oct 04 2015

my(x='x+O('x^100)); concat([0], Vec(2*x/(-1+x)/(-1+3*x))) \\ Altug Alkan, Oct 16 2015

a(n) = A000244(n) - 1.
a(n) = 2*A003462(n). - R. J. Mathar, May 01 2006
A128760(a(n)) > 0. - Reinhard Zumkeller, Mar 25 2007
G.f.: 2*x/((-1+x)*(-1+3*x)) = 1/(-1+x) - 1/(-1+3*x). - R. J. Mathar, Nov 19 2007
a(n) = Sum_{k=1..n} Sum_{m=1..k} binomial(k-1,m-1)*2^m, n >= 1. a(0)=0. From the sequence combinatorics mentioned above. Twice partial sums of powers of 3.
E.g.f.: e^(3*x) - e^x. - Mohammad K. Azarian, Jan 14 2009
a(n) = A024101(n)/A034472(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = 3*a(n-1) + 2 (with a(0)=0). - Vincenzo Librandi, Nov 19 2010
E.g.f.: -E(0) where E(k) = 1 - 3^k/(1 - x/(x - 3^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
a(n) = A227048(n,A020914(n)). - Reinhard Zumkeller, Jun 30 2013
Sum_{n>=1} 1/a(n) = A214369. - Amiram Eldar, Nov 11 2020
a(n) = Sum_{k=1..n} 2^k*binomial(n,k). - Ridouane Oudra, Jun 15 2025
From Peter Bala, Jul 01 2025: (Start)
For n >= 1, a(2*n)/a(n) = A034472(n) and a(3*n)/a(n) = A034513(n).
Modulo differences in offsets, exp( Sum_{n >= 1} a(k*n)/a(n)*x^n/n ) is the o.g.f. of A003462 (k = 2), A006100 (k = 3), A006101 (k = 4), A006102 (k = 5), A022196 (k = 6), A022197 (k = 7), A022198 (k = 8), A022199 (k = 9), A022200 (k = 10), A022201 (k = 11), A022202 (k = 12) and A022203 (k = 13).
The following are all examples of telescoping series:
Sum_{n >= 1} 3^n/(a(n)*a(n+1)) = 1/2^2; Sum_{n >= 1} 3^n/(a(n)*a(n+1)*a(n+2)) = 1/(2*8^2).
In general, for k >= 1, Sum_{n >= 1} 3^n/(a(n)*a(n+1)*...*a(n+k)) = 1/(a(1)*a(2)*...*a(k)*a(k)).
Sum_{n >= 1} 3^n/(a(n)*a(n+2)) = 5/64; Sum_{n >= 1} (-3)^n/(a(n)*a(n+2)) = -3/64.
Sum_{n >= 1} 3^n/(a(n)*a(n+4)) = 703/83200; Sum_{n >= 1} (-3)^n/(a(n)*a(n+4)) = - 417/83200. (End)

A047659 Number of ways to place 3 nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 24, 204, 1024, 3628, 10320, 25096, 54400, 107880, 199400, 348020, 579264, 926324, 1431584, 2148048, 3141120, 4490256, 6291000, 8656860, 11721600, 15641340, 20597104, 26797144, 34479744, 43915768, 55411720, 69312516, 86004800, 105919940

Offset: 0

Paul Zimmermann

Lucas mentions that the number of ways of placing p <= n non-attacking queens on an n X n chessboard is given by a polynomial in n of degree 2p and attribute the result to Mantel, professor in Delft. Cf. Stanley, exercise 15.

E. Landau, Naturwissenschaftliche Wochenschrift (Aug. 2 1896).
R. P. Stanley, Enumerative Combinatorics, vol. I, exercise 15 in chapter 4 (and its solution) asks one to show the existence of a rational generating function for the number of ways of placing k non-attacking queens on an n X n chessboard.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, Jan 26 2013. - _N. J. A. Sloane_, Feb 16 2013
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv:1609.00853 [math.CO], Sep 03 2016.
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 11.
Edmund Landau, Ueber das Achtdamenproblem und seine Verallgemeinerung, Naturwissenschaftliche Wochenschrift, Aug 02 1896.
Edouard Lucas, Récréations mathématiques, Gauthier-Villars, Paris, 1882-1894, Vol. I, p. 228.
Antal Pinter, Numerical solution of the k=3 Queens problem, 2011, P(3) at p.8-9.
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Wenxi Wang, Muhammad Usman, Alyas Almaawi, Kaiyuan Wang, Kuldeep S. Meel, Sarfraz Khurshid, A Study of Symmetry Breaking Predicates and Model Counting, National University of Singapore (2020).
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Cf. A036464, A061994, A108792, A176186, A178721.

Column k=3 of A348129.

[(3*(2*n-1)*(-1)^n +4*n^6 -40*n^5 +158*n^4 -300*n^3 +264*n^2 -86*n +3)/24: n in [0..35]]; // Vincenzo Librandi, Sep 21 2015

f:=n-> n^6/6 - 5*n^5/3 + 79*n^4/12 - 25*n^3/2 + 11*n^2 - 43*n/12 + 1/8 + (-1)^n*(n/4 - 1/8); [seq(f(n),n=1..40)]; # N. J. A. Sloane, Feb 16 2013

Table[If[EvenQ[n],n (n-2)^2 (2n^3-12n^2+23n-10)/12,(n-1)(n-3) (2n^4- 12n^3+25n^2-14n+1)/12],{n,0,30}] (* or *) LinearRecurrence[ {5,-8,0,14,-14,0,8,-5,1},{0,0,0,0,24,204,1024,3628,10320},30] (* Harvey P. Dale, Nov 06 2011 *)

a(n)=if(n%2, (n - 1)*(n - 3)*(2*n^4 - 12*n^3 + 25*n^2 - 14*n + 1), n*(n - 2)^2*(2*n^3 - 12*n^2 + 23*n - 10))/12 \\ Charles R Greathouse IV, Feb 09 2017

a(n) = n(n - 2)^2(2n^3 - 12n^2 + 23n - 10)/12 if n is even and (n - 1)(n - 3)(2n^4 - 12n^3 + 25n^2 - 14n + 1)/12 if n is odd (Landau, 1896).
a(n) = 5a(n - 1) - 8a(n - 2) + 14a(n - 4) - 14a(n - 5) + 8a(n - 7) - 5a(n - 8) + a(n - 9) for n >= 9.
G.f.: 4(9*x^4 + 35*x^3 + 49*x^2 + 21*x + 6)*x^4/((1 - x)^7*(1 + x)^2).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=24, a(5)=204, a(6)=1024, a(7)=3628, a(8)=10320, a(n) = 5*a(n-1)-8*a(n-2)+14*a(n-4)-14*a(n-5)+8*a(n-7)- 5*a(n-8)+ a(n-9). - Harvey P. Dale, Nov 06 2011
a(n) = n^6/6 - 5*n^5/3 + 79*n^4/12 - 25*n^3/2 + 11*n^2 - 43*n/12 + 1/8 + (-1)^n*(n/4 - 1/8) [Chaiken et al.]. - N. J. A. Sloane, Feb 16 2013
a(n) = (3*(2*n-1)*(-1)^n +4*n^6 -40*n^5 +158*n^4 -300*n^3 +264*n^2 -86*n +3)/24. - Antal Pinter, Oct 03 2014
E.g.f.: (exp(2*x)*(3 - 6*x^2 + 8*x^3 + 18*x^4 + 20*x^5 + 4*x^6) -3 - 6*x) / (24*exp(x)). - Vaclav Kotesovec, Feb 15 2015
For n>3, a(n) = A179058(n) -4*(n-2)*A000914(n-2) -2*(n-2)*A002415(n-1) + 2*A008911(n-1) +8*(A001752(n-4) +A007009(n-3)). - Antal Pinter, Sep 20 2015
In general, for m <= n, n >= 3, the number of ways to place 3 nonattacking queens on an m X n board is n^3/6*(m^3 - 3*m^2 + 2*m) - n^2/2*(3*m^3 - 9*m^2 + 6*m) + n/6*(2*m^4 + 20*m^3 - 77*m^2 + 58*m) - 1/24*(39*m^4 - 82*m^3 - 36*m^2 + 88*m) + 1/16*(2*m - 4*n + 1)*(1 + (-1)^(m+1)) + 1/2*(1 + abs(n - 2*m + 3) - abs(n - 2*m + 4))*(1/24*((n - 2*m + 11)^4 - 42*(n - 2*m + 11)^3 + 656*(n - 2*m + 11)^2 - 4518*(n - 2*m + 11) + 11583) - 1/16*(4*m - 2*n - 1)*(1 + (-1)^(n+1))) [Panos Louridas, idee & form 93/2007, pp. 2936-2938]. - Vaclav Kotesovec, Feb 20 2016

The formula given in the Rivin et al. paper is wrong.

Entry improved by comments from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 30 2001

A061994 Number of ways to place 4 nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 2, 82, 982, 7002, 34568, 131248, 412596, 1123832, 2739386, 6106214, 12654614, 24675650, 45704724, 80999104, 138170148, 227938788, 365106738, 569681574, 868289594, 1295775946, 1897176508, 2729909796

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

An analytical solution for the 4-queens problem permits us to combine six particular cases into a single "unified" expression: a(n) = n(n-1)(45n^6 - 855n^5 + 6945n^4 - 30891n^3 + 78864n^2 - 106226n + 53404)/1080 + (n^3 - 21/2n^2 + 28n - 14)*floor(n/2) + 32/9(n-1)*floor(n/3) + (16/9n-4)*floor((n+1)/3). The method used to derive this formula helps to fine-tune an estimate by E. Lucas for a(n) (see comment to A047659 "3-queens problem"). For any fixed value of k > 1, a(n) = n^(2k)/k! - 5/3n^(2k-1)/(k-2)! + O(n^(2k-2)). - Sergey Perepechko, Sep 16 2005

Vaclav Kotesovec, Between chessboard and computer, 1996, pp. 204-206.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Louis Azemard, Une communication de Vaclav Kotesovec, Echecs et Mathématiques, Rex Multiplex 38/1992.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 12.
Index entries for linear recurrences with constant coefficients, signature (3,1,-9,0,12,7,-15,-16,16,15,-7,-12,0,9,-1,-3,1).

Cf. A036464, A047659.

Column k=4 of A348129.

CoefficientList[Series[x^4*(2 +76*x +734*x^2 +3992*x^3 +13318*x^4 +29356*x^5 +46304*x^6 +53580*x^7 +46890*x^8 +29768*x^9 +13522*x^10 +3804*x^11 +574*x^12)/((1-x)^3*(1-x^2)^4*(1-x^3)^2), {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)
LinearRecurrence[{3,1,-9,0,12,7,-15,-16,16,15,-7,-12,0,9,-1,-3,1}, {0,0,0,0,2,82, 982,7002,34568,131248,412596,1123832,2739386,6106214,12654614,24675650, 45704724}, 40] (* Harvey P. Dale, Jan 21 2017 *)

def p(x): return x^4*(2 +76*x +734*x^2 +3992*x^3 +13318*x^4 +29356*x^5 +46304*x^6 +53580*x^7 +46890*x^8 +29768*x^9 +13522*x^10 +3804*x^11 +574*x^12)/((1-x)^3*(1-x^2)^4*(1-x^3)^2)
def A061994_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( p(x) ).list()
A061994_list(40) # G. C. Greubel, Apr 30 2022

G.f.: x^4*(2 + 76*x + 734*x^2 + 3992*x^3 + 13318*x^4 + 29356*x^5 + + 46304*x^6 + + 53580*x^7 + 46890*x^8 + 29768*x^9 + 13522*x^10 + 3804*x^11 + 574*x^12)/((1-x)^3*(1-x^2)^4*(1-x^3)^2).
Recurrence: a(n) = 3*a(n-1) + a(n-2) - 9*a(n-3) + 12*a(n-5) + 7*a(n-6) - 15*a(n-7) - 16*a(n-8) + 16*a(n-9) + 15*a(n-10) - 7*a(n-11) - 12*a(n-12) + 9*a(n-14) - a(n-15) - 3*a(n-16) + a(n-17), n >= 17.
Explicit formula (V. Kotesovec, 1992) for n >= 2: a(n) = n^8/24 - 5*n^7/6 + 65*n^6/9 - 1051*n^5/30 + 817*n^4/8 added to one of the following terms:
  - 4769*n^3/27 + 1963*n^2/12 - 1769*n/30         if n = 0 (mod 6)
  - 9565*n^3/54 + 1013*n^2/6 - 6727*n/90 + 257/27 if n = 1 (mod 6)
  - 4769*n^3/27 + 1963*n^2/12 - 5467*n/90 + 28/27 if n = 2 (mod 6)
  - 9565*n^3/54 + 1013*n^2/6 - 2189*n/30 + 7      if n = 3 (mod 6)
  - 4769*n^3/27 + 1963*n^2/12 - 5467*n/90 + 68/27 if n = 4 (mod 6)
  - 9565*n^3/54 + 1013*n^2/6 - 6727*n/90 + 217/27 if n = 5 (mod 6).
a(n) = n^8/24 - 5n^7/6 + 65n^6/9 - 1051n^5/30 + 817n^4/8 - 19103n^3/108 + 3989n^2/24 - 18131n/270 + 253/54 + (n^3/4 - 21n^2/8 + 7n - 7/2)*(-1)^n + 32*(n - 1)/27*cos(2*Pi*n/3) + 40/81*sqrt(3)*sin(2*Pi*n/3). - Vaclav Kotesovec, Feb 11 2010
E.g.f.: (3*(exp(2*x)*(5060 - 4645*x + 1755*x^2 - 590*x^3 + 480*x^4 + 414*x^5 + 870*x^6 + 360*x^7 + 45*x^8) - 135*(28 + 37*x + 15*x^2 + 2*x^3)) - 1920 * exp(x/2) * (2+x) * cos(sqrt(3)*x/2) - 320 * sqrt(3) * exp(x/2) * (6*x-5) * sin(sqrt(3)*x/2)) / (3240 * exp(x)). - Vaclav Kotesovec, Feb 15 2015

Minor edits by Vaclav Kotesovec, Feb 15 2015

A172123 Number of ways to place 2 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 4, 26, 92, 240, 520, 994, 1736, 2832, 4380, 6490, 9284, 12896, 17472, 23170, 30160, 38624, 48756, 60762, 74860, 91280, 110264, 132066, 156952, 185200, 217100, 252954, 293076, 337792, 387440, 442370, 502944, 569536, 642532

Offset: 1

Vaclav Kotesovec, Jan 26 2010

E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A036464.

[n*(n-1)*(3*n^2-n+2)/6: n in [1..40]]; // Vincenzo Librandi, Apr 29 2013

I:=[0, 4, 26, 92, 240]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 29 2013

CoefficientList[Series[-2 x (x+1)(x+2)/(x-1)^5, {x, 0, 80}], x] (* Vincenzo Librandi, Apr 29 2013 *)

a(n)=n*(n-1)*(3*n^2-n+2)/6 \\ Charles R Greathouse IV, Jun 16 2015

a(n) = n*(n - 1)*(3*n^2 - n + 2)/6.
G.f.: -2*x^2*(x+1)*(x+2)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Vincenzo Librandi, Apr 29 2013

A172225 Number of ways to place 2 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 2, 24, 96, 260, 570, 1092, 1904, 3096, 4770, 7040, 10032, 13884, 18746, 24780, 32160, 41072, 51714, 64296, 79040, 96180, 115962, 138644, 164496, 193800, 226850, 263952, 305424, 351596, 402810, 459420, 521792, 590304

Offset: 1

Vaclav Kotesovec, Jan 29 2010

A wazir is a (fairy chess) leaper [0,1].

Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p. 829.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A036464, A061995, A172132, A172123, A172137.

I:=[0, 2, 24, 96, 260]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[n*(n-1)*(n^2+n-4)/2: n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

Table[n (n - 1) (n^2 + n - 4) / 2, {n, 40}] (* Vincenzo Librandi, Apr 30 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,24,96,260},40] (* Harvey P. Dale, Jun 04 2023 *)

Explicit formula (Christian Poisson, 1990): a(n) = n*(n-1)*(n^2+n-4)/2.
G.f.: 2*x^2*(2*x^2-7*x-1)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Apr 30 2013
a(n) = 2*A239352(n). - R. J. Mathar, Jan 09 2018
a(n) = A232833(n,2). - R. J. Mathar, Apr 11 2024

A172132 Number of ways to place 2 nonattacking knights on an n X n board.

Original entry on oeis.org

0, 6, 28, 96, 252, 550, 1056, 1848, 3016, 4662, 6900, 9856, 13668, 18486, 24472, 31800, 40656, 51238, 63756, 78432, 95500, 115206, 137808, 163576, 192792, 225750, 262756, 304128, 350196, 401302, 457800, 520056, 588448, 663366

Offset: 1

Vaclav Kotesovec, Jan 26 2010

E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A036464, A172123.

Column k=2 of A244081.

I:=[0, 6, 28, 96, 252]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[(n-1)*(n+4)*(n^2-3*n+4)/2: n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

Table[(n-1)(n+4)(n^2 -3n +4)/2, {n, 40}] (* Vincenzo Librandi, Apr 30 2013 *)

[(n-1)*(n+4)*(n^2-3*n+4)/2 for n in (1..40)] # G. C. Greubel, Apr 18 2022

a(n) = (n - 1)*(n + 4)*(n^2 - 3*n + 4)/2.
G.f.: 2*(12*x^4-39*x^3+37*x^2-20*x+4)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Vincenzo Librandi, Apr 30 2013
E.g.f.: (1/2)*(16 + (-16 + 16*x - 2*x^2 + 6*x^3 + x^4)*exp(x)). - G. C. Greubel, Apr 18 2022

A348129 Number T(n,k) of ways to place k nonattacking queens on an n X n board; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 0, 1, 9, 8, 0, 1, 16, 44, 24, 2, 1, 25, 140, 204, 82, 10, 1, 36, 340, 1024, 982, 248, 4, 1, 49, 700, 3628, 7002, 4618, 832, 40, 1, 64, 1288, 10320, 34568, 46736, 22708, 3192, 92, 1, 81, 2184, 25096, 131248, 310496, 312956, 119180, 13848, 352, 1, 100, 3480, 54400, 412596, 1535440, 2716096, 2119176, 636524, 56832, 724

Offset: 0

Alois P. Heinz, Oct 01 2021

			T(3,2) = 8:
  .-----. .-----. .-----. .-----. .-----. .-----. .-----. .-----.
  |Q . .| |Q . .| |. . Q| |. . Q| |. . .| |. Q .| |. Q .| |. . .|
  |. . Q| |. . .| |. . .| |Q . .| |Q . .| |. . .| |. . .| |. . Q|
  |. . .| |. Q .| |. Q .| |. . .| |. . Q| |. . Q| |Q . .| |Q . .|
  `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,    0;
  1,  9,    8,     0;
  1, 16,   44,    24,      2;
  1, 25,  140,   204,     82,     10;
  1, 36,  340,  1024,    982,    248,      4;
  1, 49,  700,  3628,   7002,   4618,    832,     40;
  1, 64, 1288, 10320,  34568,  46736,  22708,   3192,    92;
  1, 81, 2184, 25096, 131248, 310496, 312956, 119180, 13848, 352;
  ...

Columns k=0-8 give: A000012, A000290, A036464, A047659, A061994, A108792, A176186, A178721, A252593.
Main diagonal gives A000170.
Row sums give A287227.
T(2n,n) gives A348130.
Cf. A090642, A144084, A178717.

A172517 Number of ways to place 2 nonattacking queens on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 32, 100, 288, 588, 1152, 1944, 3200, 4840, 7200, 10140, 14112, 18900, 25088, 32368, 41472, 51984, 64800, 79380, 96800, 116380, 139392, 165000, 194688, 227448, 264992, 306124, 352800, 403620, 460800, 522720, 591872, 666400, 749088, 837828

Offset: 1

Vaclav Kotesovec, Feb 05 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A007705, A036464.

CoefficientList[Series[- 4 x^3 (x^3 + 6 x^2 + 9 x + 8) / ((x - 1)^5 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,0,0,32,100,288,588,1152},40] (* Harvey P. Dale, Sep 22 2015 *)

a(n) = n^2*(n-2)^2/2 if n is even and a(n) = n^2*(n-1)(n-3)/2 if n is odd.
G.f.: -4*x^4*(x^3+6*x^2+9*x+8) / ((x-1)^5*(x+1)^3). - Colin Barker, Jan 09 2013
a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8). - Wesley Ivan Hurt, May 28 2021

A172141 Number of ways to place 2 nonattacking nightriders on an n X n board.

Original entry on oeis.org

0, 6, 28, 96, 240, 518, 980, 1712, 2784, 4310, 6380, 9136, 12688, 17206, 22820, 29728, 38080, 48102, 59964, 73920, 90160, 108966, 130548, 155216, 183200, 214838, 250380, 290192, 334544, 383830, 438340, 498496, 564608, 637126

Offset: 1

Vaclav Kotesovec, Jan 26 2010

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

Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A036464, A172123, A172132, A172137.

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)): n in [1..40]]; // G. C. Greubel, Apr 21 2022

CoefficientList[Series[2*x*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2), {x,0,40}], x] (* Vincenzo Librandi, May 27 2013 *)

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)) for n in (1..40)] # G. C. Greubel, Apr 21 2022

Explicit formula (Christian Poisson, 1990): a(n) = n*(3*n^3 - 5*n^2 + 9*n - 4)/6 if n is even and a(n) = n*(n - 1)*(3*n^2 - 2*n + 7)/6 if n is odd.
G.f.: 2*x^2*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2). - Vaclav Kotesovec, Mar 25 2010
From G. C. Greubel, Apr 21 2022: (Start)
a(n) = (1/12)*n*(3*(-1)^n - (11 - 18*n + 10*n^2 - 6*n^3)).
E.g.f.: (x/12)*(-3*exp(-x) + (3 + 30*x + 26*x^2 + 6*x^3)exp(x)). (End)

cp's OEIS Frontend

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

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A024023 a(n) = 3^n - 1.

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A047659 Number of ways to place 3 nonattacking queens on an n X n board.

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A061994 Number of ways to place 4 nonattacking queens on an n X n board.

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A172123 Number of ways to place 2 nonattacking bishops on an n X n board.

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A172225 Number of ways to place 2 nonattacking wazirs on an n X n board.

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