A099152 -id:A099152 - 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

A187235 Number of ways to place n nonattacking semi-bishops on an n X n board.

Original entry on oeis.org

1, 5, 51, 769, 15345, 381065, 11323991, 391861841, 15476988033, 687029386845, 33861652925595, 1834814222811361, 108411291759763681, 6936921762461326545, 477881176664541171375, 35264213540563039871265, 2775185864375851234241985, 232010235620834821000259765, 20534530616200868936398461635

Offset: 1

Vaclav Kotesovec, Mar 08 2011

Two semi-bishops do not attack each other if they are in the same NorthWest-SouthEast diagonal.

Conjecture: Number of parity preserving permutations of the set {1, 2, ..., 2n+1} with exactly n+1 cycles (see A246117). - Peter Luschny, Feb 09 2015

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 260-265.

Cf. A238261, A238262, A002465, A099152, A000255, A129256.

Table[If[n==1,1,Coefficient[Expand[Product[x+i,{i,1,n}]*Product[x+i,{i,1,n-1}],x],x,n-1]],{n,1,50}]
Table[(-1)^n*Sum[StirlingS1[n+1,j]*StirlingS1[n,n-j+1],{j,1,n}],{n,1,50}] (* Explicit formula, Vaclav Kotesovec, Mar 24 2011 *)

a(n) = {(-1)^n*sum(i=0, n, stirling(n,i,1) * stirling(n+1,n-i+1,1))} \\ Andrew Howroyd, May 09 2020

a(n)/(n-1)! ~ 0.24252191 * 4.9108149^n where the second constant is 1/(z*(1-z)) = 4.910814964..., where z=0.715331862959... is a root of the equation z=2*(z-1)*log(1-z).
For constants see A238261 and A238262. - Vaclav Kotesovec, Feb 21 2014
a(n) = (-1)^n * Sum_{i=0..n} Stirling1(n,i) * Stirling1(n+1,n-i+1). - Ryan Brooks, May 09 2020

A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

Original entry on oeis.org

0, 1, 13, 58, 170, 395, 791, 1428, 2388, 3765, 5665, 8206, 11518, 15743, 21035, 27560, 35496, 45033, 56373, 69730, 85330, 103411, 124223, 148028, 175100, 205725, 240201, 278838, 321958, 369895, 422995, 481616, 546128, 616913, 694365, 778890

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

Row sums of A103219.
From Bruno Berselli, Dec 10 2010: (Start)
a(n) = n*A002412(n) - Sum_{i=0..n-1} A002412(i). More generally: n^2*(n+1)*(2*d*n-2*d+3)/6 - (Sum_{i=0..n-1} i*(i+1)*(2*d*i-2*d+3))/6 = n * (n+1) * (3*d*n^2-d*n+4*n-2*d+2)/12; in this sequence is d=2.
The inverse binomial transform yields 0, 1, 11, 22, 12, 0, 0 (0 continued). (End)
a(n-1) is also number of ways to place 2 nonattacking semi-queens (see A099152) on an n X n board. - Vaclav Kotesovec, Dec 22 2011
Also, one-half the even-indexed terms of the partial sums of A045947. - J. M. Bergot, Apr 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002412, A002418, A039755, A099152, A103219, A015237 (first diffs), A024196.

for(n=0,100,print1((3*n^4+4*n^3-n)/6,","))

CoefficientList[Series[- x (1 + 8 x + 3 x^2) / (x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,13,58,170},40] (* Harvey P. Dale, Jan 23 2016 *)

a(n)=n*(n+1)*(3*n^2+n-1)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+8*x+3*x^2)/(1-x)^5.
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013
a(n) = a(n-1) + (2*n-1)*n^2 with a(0)=0, see A015237. - J. M. Bergot, Jun 10 2017
From Wesley Ivan Hurt, Nov 20 2021: (Start)
a(n) = Sum_{k=1..n} k * C(2*k,2).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). (End)
From Peter Bala, Sep 03 2023: (Start)
a(n) = Sum_{1 <= i <= j <= n} (2*i - 1)*(2*j - 1).
Second subdiagonal of A039755. (End)

A088789 E.g.f.: REVERT(2x/(1+exp(x))) = Sum_{n>=0} a(n)x^n/n!.

Original entry on oeis.org

0, 1, 1, 3, 14, 90, 738, 7364, 86608, 1173240, 17990600, 308055528, 5826331440, 120629547584, 2713659864832, 65909241461760, 1718947213795328, 47912968352783232, 1421417290991105664, 44717945211445216640, 1487040748881346835200, 52117255681017313721088

Offset: 0

Paul D. Hanna, Oct 15 2003

a(n+1) is also number of ways to place n nonattacking composite pieces semi-rook + semi-bishop on an n X n board. Two semi-bishops (see A187235) do not attack each other if they are in the same northwest-southeast diagonal. Two semi-rooks do not attack each other if they are in the same column (see also semi-queens, A099152). - Vaclav Kotesovec, Dec 22 2011

Alois P. Heinz, Table of n, a(n) for n = 0..200
V. Dotsenko, Pattern avoidance in labelled trees, arXiv preprint arXiv:1110.0844 [math.CO], 2011.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 716-719.
R. W. Whitty, Rook polynomials on two-dimensional surfaces and graceful labellings of graphs, Discrete Math., 308 (2008), 674-683.

Main diagonal of A378561 (shifted).

a:= n->coeff(series(x/2-LambertW(-1/2*x*exp(1/2*x)), x=0, n+1), x, n)*n!:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 14 2008

Table[n!/2^n*Sum[2^j/j!*StirlingS2[n-1,n-j],{j,1,n}],{n,1,50}] (* Vaclav Kotesovec, Dec 25 2011 *)
With[{nmax = 50}, CoefficientList[Series[x/2 - LambertW[-x*Exp[x/2]/2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)

a(n)=local(A); if(n<0,0,A=x+O(x^n);n!*polcoeff(serreverse(2*x/(1 + exp(x))), n))

x='x+O('x^50); concat([0], Vec(serlaplace(x/2 - lambertw(-x*exp(x/2)/2)))) \\ G. C. Greubel, Nov 14 2017

E.g.f.: x/2 - LambertW(-x*exp(x/2)/2). - Vladeta Jovovic, Feb 12 2008
a(n) = (1/2^n)*Sum_{k=1..n} binomial(n,k)*k^(n-1) = A038049(n)/2^n, n>1. - Vladeta Jovovic, Feb 12 2008
Asymptotics: a(n)/(n-2)! ~ b * q^(n-1) * sqrt(n), where q = 1/(2*LambertW(1/exp(1))) = 1.795560738334311... is the root of the equation 2*q = exp(1+1/(2*q)) and b = 1/(2*LambertW(1/exp(1))) * sqrt((1+LambertW(1/exp(1)))/(2*Pi)) = 0.8099431005... - Vaclav Kotesovec, Dec 22 2011, updated Sep 25 2012

More terms from Alois P. Heinz, Aug 14 2008

Minor edits by Vaclav Kotesovec, Mar 31 2014

A125182 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that the set {p(i)-i, i=1,2,...,n} has exactly k elements (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 12, 7, 1, 4, 38, 54, 23, 1, 8, 77, 248, 303, 83, 1, 6, 160, 824, 2008, 1636, 405, 1, 11, 285, 2320, 9449, 15789, 10352, 2113, 1, 10, 476, 5564, 37237, 102726, 133293, 70916, 12657, 1, 14, 799, 13172, 122708, 536900, 1158368, 1177168, 537373, 82297

Offset: 1

Emeric Deutsch, Nov 24 2006

Row sums are the factorial numbers (A000142). T(n,1)=1 (the identity permutation). T(n,2) = A065608(n) = (sum of divisors of n)-(number of divisors of n). T(n,n) = A099152(n). In the first Maple program define n (<=10) to obtain row n.

T(n,k) is also the number of permutations p of {1,2,...,n} such that the set {p(i) + i, i=1,2,...,n} has exactly k elements (1<=k<=n). Example: T(4,2)=4 because we have 1432, 3412, 2143 and 3214. - Emeric Deutsch, Nov 28 2008

			T(4,2) = 4 because we have 4123, 3412, 2143 and 2341.
Triangle starts:
  1;
  1, 1;
  1, 2,  3;
  1, 4, 12,  7;
  1, 4, 38, 54, 23;
  ...

Alois P. Heinz, Rows n = 1..12, flattened
M. Alekseyev, E. Deutsch, and J. H. Steelman, Problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465. - _Emeric Deutsch_, Apr 23 2009

Cf. A000142, A065608, A099152, A125183, A306455.

n:=7: with(combinat): P:=permute(n): for j from 1 to n! do c[j]:=0 od: for j from 1 to n! do if nops({seq(P[j][i]-i,i=1..n)}) = 1 then c[1]:=c[1]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 2 then c[2]:=c[2]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 3 then c[3]:=c[3]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 4 then c[4]:=c[4]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 5 then c[5]:=c[5]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 6 then c[6]:=c[6]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 7 then c[7]:=c[7]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 8 then c[8]:=c[8]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 9 then c[9]:=c[9]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 10 then c[10]:=c[10]+1 else fi od: seq(c[i],i=1..n);
# second Maple program:
b:= proc(p, s) option remember; `if`(p={}, x^nops(s),
      add(b(p minus {t}, s union {t+nops(p)}), t=p))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b({$1..n}, {})):
seq(T(n), n=1..9);  # Alois P. Heinz, May 04 2014; revised, Sep 08 2018

b[i_, p_List, s_List] := b[i, p, s] = If[p == {}, x^Length[s], Sum[b[i+1, p ~Complement~ {t}, s ~Union~ {t+i}], {t, p}]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 1, n}]][b[1, Range[n], {}]]; Table[T[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A306455(n). - Alois P. Heinz, Feb 16 2019

A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 3, 52, 370, 1620, 5285, 14168, 33012, 69240, 133815, 242220, 415558, 681772, 1076985, 1646960, 2448680, 3552048, 5041707, 7018980, 9603930, 12937540, 17184013, 22533192, 29203100, 37442600, 47534175, 59796828, 74589102, 92312220, 113413345, 138388960

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A099152, A047659, A103220, A202655, A202656, A202657.

Rest@ CoefficientList[Series[-x^3*(17 x^3 + 69 x^2 + 31 x + 3)/(x - 1)^7, {x, 0, 32}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = 1/6*(n-2)*(n-1)*n*(n^3-5*n^2+8*n-3).

G.f.: -x^3*(17*x^3 + 69*x^2 + 31*x + 3)/(x-1)^7.

A202655 Number of ways to place 4 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 7, 223, 2429, 15045, 66122, 230074, 675798, 1745318, 4073993, 8764753, 17630795, 33522531, 60756612, 105666148, 177293340, 288246972, 455749371, 702898611, 1060173961, 1567213681, 2274896558, 3247759614, 4566786770, 6332604226, 8669120733, 11727651845

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7,-19,21,6,-42,42,-6,-21,19,-7,1).

Cf. A099152, A061994, A103220, A202654, A202656, A202657.

Rest@ CoefficientList[Series[-x^4*(151 x^6 + 1022 x^5 + 2233 x^4 + 2132 x^3 + 1001 x^2 + 174 x + 7)/((x - 1)^9*(x + 1)^2), {x, 0, 29}], x] (* Michael De Vlieger, Aug 19 2019 *)
LinearRecurrence[{7,-19,21,6,-42,42,-6,-21,19,-7,1},{0,0,0,7,223,2429,15045,66122,230074,675798,1745318},30] (* Harvey P. Dale, Mar 17 2025 *)

a(n) = n^8/24 - 2*n^7/3 + 41*n^6/9 - 257*n^5/15 + 341*n^4/9 - 97*n^3/2 + 2341*n^2/72 - 87*n/10 + (n/2 - 1/2)*floor(n/2).

G.f.: -x^4*(151*x^6 + 1022*x^5 + 2233*x^4 + 2132*x^3 + 1001*x^2 + 174*x + 7)/((x-1)^9*(x+1)^2).

A202656 Number of ways to place 5 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 23, 1104, 16945, 141696, 810746, 3568352, 12948318, 40514560, 112720393, 285073712, 666143975, 1456288512, 3007576740, 5913372864, 11138305068, 20202100224, 35433809451, 60316600080, 99947225741, 161638967424, 255701773822, 396439174560, 603407582570

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -17, 7, 43, -77, 11, 99, -99, -11, 77, -43, -7, 17, -7, 1).

Cf. A099152, A108792, A103220, A202654, A202655, A202657.

Rest@ CoefficientList[Series[-x^5*(1899 x^9 + 16515 x^8 + 60512 x^7 + 116784 x^6 + 137646 x^5 + 98222 x^4 + 41688 x^3 + 9608 x^2 + 943 x + 23)/((x - 1)^11*(x + 1)^4), {x, 0, 27}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = n^10/120 - 2*n^9/9 + 95*n^8/36 - 183*n^7/10 + 14663*n^6/180 - 1201*n^5/5 + 16753*n^4/36 - 25364*n^3/45 + 68293*n^2/180 - 12781*n/120 + (n^3/2 - 6*n^2 + 39*n/2 - 61/4)*floor(n/2).

G.f.: -x^5*(1899*x^9 + 16515*x^8 + 60512*x^7 + 116784*x^6 + 137646*x^5 + 98222*x^4 + 41688*x^3 + 9608*x^2 + 943*x + 23)/((x-1)^11*(x+1)^4).

A202657 Number of ways to place 6 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 83, 6107, 126376, 1377328, 9984758, 54399330, 239675936, 895773148, 2935757573, 8641608781, 23259768860, 58039719112, 135720432200, 299995484600, 631220344328, 1271607596876, 2464466665667, 4613731163831, 8372196591052, 14769606793684, 25395151577010

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (5, -4, -18, 30, 24, -61, -31, 74, 70, -92, -104, 104, 92, -70, -74, 31, 61, -24, -30, 18, 4, -5, 1).

Cf. A099152, A176186, A103220, A202654, A202655, A202656.

Rest@ CoefficientList[Series[-x^6*(31709 x^16 + 377288 x^15 + 2265487 x^14 + 8441426 x^13 + 22166758 x^12 + 43217858 x^11 + 64805639 x^10 + 75943200 x^9 + 70077016 x^8 + 50738668 x^7 + 28477437 x^6 + 12074418 x^5 + 3711058 x^4 + 771370 x^3 + 96173 x^2 + 5692 x + 83)/((x - 1)^13*(x + 1)^6*(x^2 + x + 1)^2), {x, 0, 26}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = n^12/720 - n^11/18 + 73*n^10/72 - 72247*n^9/6480 + 5909*n^8/72 - 320653*n^7/756 + 112795*n^6/72 - 8892919*n^5/2160 + 8086231*n^4/1080 - 5740271*n^3/648 + 2598425*n^2/432 - 13161367*n/7560 + (n^5/4 - 77*n^4/12 + 757*n^3/12 - 7007*n^2/24 + 14581*n/24 - 1677/4)*floor(n/2) + (64*n/9 - 88/9)*floor(n/3) + (8*n/3 - 52/9)*floor((n + 1)/3).

G.f.: -x^6*(31709*x^16 + 377288*x^15 + 2265487*x^14 + 8441426*x^13 + 22166758*x^12 + 43217858*x^11 + 64805639*x^10 + 75943200*x^9 + 70077016*x^8 + 50738668*x^7 + 28477437*x^6 + 12074418*x^5 + 3711058*x^4 + 771370*x^3 + 96173*x^2 + 5692*x + 83)/((x-1)^13*(x+1)^6*(x^2+x+1)^2).

A189843 Number of ways to place n nonattacking composite pieces rook + semi-rider[2,2] on an n X n chessboard.

Original entry on oeis.org

1, 2, 5, 18, 71, 356, 2097, 14212, 105821, 887576, 8093601, 81310936, 876456695, 10257217440, 127631146697, 1705775408656

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+2k)-p(j)<>2k for all j>=1, k>=1, j+2k<=n

about semi-pieces see semi-bishop (A187235) and semi-queen (A099152)

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
Wikipedia, Fairy chess piece

Cf. A189281, A099152, A189838, A187235

cp's OEIS Frontend

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

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A187235 Number of ways to place n nonattacking semi-bishops on an n X n board.

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A103220 a(n) = n*(n+1)*(3*n^2+n-1)/6.

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A088789 E.g.f.: REVERT(2*x/(1+exp(x))) = Sum_{n>=0} a(n)*x^n/n!.

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A125182 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that the set {p(i)-i, i=1,2,...,n} has exactly k elements (1<=k<=n).

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A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

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A202655 Number of ways to place 4 nonattacking semi-queens on an n X n board.

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A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

A088789 E.g.f.: REVERT(2x/(1+exp(x))) = Sum_{n>=0} a(n)x^n/n!.