A051224 -id:A051224 - cp's OEIS frontend

A051223 Number of ways of placing n nonattacking superqueens on an n X n board.

1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 44, 156, 1876, 5180, 32516, 202900, 1330622, 8924976, 64492432, 495864256, 3977841852, 34092182276, 306819842212, 2883202816808, 28144109776812, 286022102245804

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

A superqueen moves like a queen and a knight.
A linear-time algorithm giving an explicit solution for any n >= 10 for the n-super-queens-problem can be found at the link. Included is an online solver, implemented in JavaScript. - Frank Schwellinger (nummer_eins(AT)web.de), Mar 19 2004. [But see the next comment - N. J. A. Sloane, Jul 01 2021]
Escamocher and O'Sullivan (2021) claim Schwellinger's algorithm is incorrect, and that their own algorithm is the first published linear-time algorithm. - N. J. A. Sloane, Jul 01 2021

D. Bill, Durango Bill's The N-Queens Problem
Guillaume Escamocher and Barry O'Sullivan, Leprechauns on the chessboard, Discrete Mathematics, 344(5), 2021. See especially page 3.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed.
R. Oprisch, N x N SuperQueens Solutions Table
F. Schwellinger, Explicit Solution To the N-Super-Queens Problem.
W. Schubert, N-Queens page

Cf. A051224, A000170.

a(20) from Bill link added Jul 25 2006
a(21)-a(23) from R. Oprisch's website added by Max Alekseyev, Sep 29 2006
a(24)-a(26) from W. Schubert, Jul 31 2009, Nov 29 2009, Jan 18 2011

A102388 Number of ways of placing n nonattacking Queens of the Night on an n X n board.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 44, 6, 78, 8, 16, 18, 234, 124, 468, 516, 882, 2092, 7068, 22794, 85456, 275732, 974048, 3698242, 14120996, 59531852, 252272512, 1163430462, 5229335374

Offset: 1

Stefan Wernli, Peter Syski (swernli(AT)fas.harvard.edu), Jan 07 2005

A Queen of the Night can move like a Queen or a Nightrider, which is a rider along straight lines of Knight moves.

Noam D. Elkies, Chess Pages and Links.
P. Syski, S. J. Wernli, Queen of the Night on an n X n board.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 350.
W. Schubert, N-Queens page.

Cf. A051223, A051224, A000170.

Terms a(20)-a(28) from Vaclav Kotesovec, Jun 18 2010 and Feb 02 2011
Terms a(29)-a(32) from Wolfram Schubert, Jul 24 2011
Term a(33) from Wolfram Schubert, May 27 2012

A172200 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 20, 92, 260, 580, 1120, 1960, 3192, 4920, 7260, 10340, 14300, 19292, 25480, 33040, 42160, 53040, 65892, 80940, 98420, 118580, 141680, 167992, 197800, 231400, 269100, 311220, 358092, 410060, 467480, 530720, 600160, 676192, 759220, 849660

Offset: 1

Vaclav Kotesovec, Jan 29 2010

A amazon (superqueen) moves like a queen and a knight.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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 (5,-10,10,-5,1).

Cf. A036464, A051223, A051224.

[(n-1)*(n-2)*(n-3)*(3*n+8)/6: n in [1..50]]; // Vincenzo Librandi, May 27 2013

CoefficientList[Series[4x^3(5-2x)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,0,20,92},40] (* or *) Table[(n-1)(n-2)(n-3)(3n+8)/6,{n,40}] (* Harvey P. Dale, May 16 2021 *)

[binomial(n-1,3)*(3*n+8) for n in (1..50)] # G. C. Greubel, Apr 28 2022

Explicit formula (Christian Poisson, 1990): a(n) = (n - 1)(n - 2)(n - 3)(3n + 8)/6.
G.f.: 4*x^4*(5-2*x)/(1-x)^5. - Colin Barker, Jan 09 2013
E.g.f.: 8 + (1/6)*(-48 +48*x -24*x^2 +8*x^3 +3*x^4)*exp(x). - G. C. Greubel, Apr 28 2022

A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 48, 424, 1976, 6616, 17852, 41544, 86660, 166288, 298616, 508200, 827168, 1296744, 1968676, 2907016, 4189772, 5910944, 8182400, 11136168, 14926536, 19732600, 25760588, 33246664, 42459476, 53703216, 67320392, 83695144

Offset: 1

Vaclav Kotesovec, Jan 29 2010

An amazon (superqueen) moves like a queen and a knight.

Panos Louridas, idee & form 93/2007, pp. 2936-2938.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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 (5,-8,0,14,-14,0,8,-5,1).

Cf. A047659, A051223, A051224, A061989, A172200.

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7 ))); // G. C. Greubel, Apr 29 2022

CoefficientList[Series[4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7), {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

[(1/24)*(4*n^6 -40*n^5 +62*n^4 +628*n^3 -2904*n^2 +4074*n -1341 +3*(-1)^n*(2*n-1)) -4*(5*bool(n==1) +2*bool(n==2) -bool(n==3)) for n in (1..40)] # G. C. Greubel, Apr 29 2022

Explicit formula (Panos Louridas, 2007): a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2040*n - 672)/12 if n is even (n >= 4) and a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2034*n - 669)/12 if n is odd (n >= 5).
G.f.: 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7). - Vaclav Kotesovec, Mar 24 2010
a(n) = (1/24)*(4*n^6 - 40*n^5 + 62*n^4 + 628*n^3 - 2904*n^2 + 4074*n - 1341 + 3*(-1)^n*(2*n-1)) - 20*[n=1] - 8*[n=2] + 4*[n=3]. - G. C. Greubel, Apr 29 2022

A174642 Number of ways to place 4 nonattacking amazons (superqueens) on a 4 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 12, 60, 180, 432, 900, 1692, 2940, 4800, 7452, 11100, 15972, 22320, 30420, 40572, 53100, 68352, 86700, 108540, 134292, 164400, 199332, 239580, 285660, 338112, 397500, 464412, 539460, 623280, 716532, 819900, 934092, 1059840

Offset: 1

Vaclav Kotesovec, Mar 25 2010

An amazon (superqueen) moves like a queen and a knight

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
The Oprisch Family Web Site
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A173214, A172201, A172200, A051223, A051224, A036464.

CoefficientList[Series[- 12 x^7 (x^3 + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

G.f.: -12*x^8*(x^3+1)/(x-1)^5.

Explicit formula: a(n) = (n-7)(n^3-21n^2+158n-420), n>=7.

More terms from Vincenzo Librandi, May 30 2013

A352661 Number of doubly symmetric characteristic solutions to the n-superqueens problem.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 12, 17, 0, 0, 60, 82

Offset: 1

Don Knuth, Mar 25 2022

Superqueens are also called amazons. They combine the moves of queen and knight.

			For n=12 the a(12)=2 solutions are
  +-------------------------+ +-------------------------+
  | . . . . A . . . . . . . | | . . . . A . . . . . . . |
  | . . . . . . . . . A . . | | . . . . . . . . . A . . |
  | . A . . . . . . . . . . | | . A . . . . . . . . . . |
  | . . . . . A . . . . . . | | . . . . . . A . . . . . |
  | . . . . . . . . . . . A | | . . . . . . . . . . . A |
  | . . . . . . . . A . . . | | . . . A . . . . . . . . |
  | . . . A . . . . . . . . | | . . . . . . . . A . . . |
  | A . . . . . . . . . . . | | A . . . . . . . . . . . |
  | . . . . . . A . . . . . | | . . . . . A . . . . . . |
  | . . . . . . . . . . A . | | . . . . . . . . . . A . |
  | . . A . . . . . . . . . | | . . A . . . . . . . . . |
  | . . . . . . . A . . . . | | . . . . . . . A . . . . |
  +-------------------------+ +-------------------------+

Martin Gardner, Fractal Music, Hypercards, and More, W H Freeman, 1991, page 238 (based on his column in Scientific American, June 1979).
D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3 (draft, March 2022).

A051224 = A352661 + A352662 + A352663.

A051223 = 2*A352661 + 4*A352662 + 8*A352663 (when n>1).

A352662 Number of singly symmetric characteristic solutions to the n-superqueens problem.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 11, 49, 79, 245, 498, 1192, 3798, 11594

Offset: 1

Don Knuth, Mar 25 2022

Superqueens are also called amazons. They combine the moves of queen and knight.

			For n=10 the a(10)=1 solution is
  +---------------------+
  | . . . A . . . . . . |
  | . . . . . . . A . . |
  | A . . . . . . . . . |
  | . . . . A . . . . . |
  | . . . . . . . . A . |
  | . A . . . . . . . . |
  | . . . . . A . . . . |
  | . . . . . . . . . A |
  | . . A . . . . . . . |
  | . . . . . . A . . . |
  +---------------------+

D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3 (draft, March 2022).

Cf. A051223, A051224, A352661, A352663.

A352663 Number of asymmetric characteristic solutions to the n-superqueens problem.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 18, 231, 642, 4040, 25320, 166201, 1115373, 8060958, 61981118, 497224414

Offset: 1

Don Knuth, Mar 25 2022

			One of the a(11)=5 solutions is
  +-----------------------+
  | A . . . . . . . . . . |
  | . . . . A . . . . . . |
  | . . . . . . . . A . . |
  | . A . . . . . . . . . |
  | . . . . . A . . . . . |
  | . . . . . . . . . A . |
  | . . A . . . . . . . . |
  | . . . . . . A . . . . |
  | . . . . . . . . . . A |
  | . . . A . . . . . . . |
  | . . . . . . . A . . . |
  +-----------------------+
and the other four are obtained by wraparound shifts.

D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3 (draft, March 2022).

Cf. A051223, A051224, A352661, A352662.

A007631 Number of solutions to non-attacking reflecting queens problem.

Original entry on oeis.org

1, 1, 0, 0, 2, 4, 0, 2, 10, 32, 38, 140, 496, 1186, 3178, 16792, 82038, 289566, 1139874, 5914118, 33800010, 142337180, 721286448, 4384569864

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

a(n) is the number of ways to pair the natural numbers from 1 to n with those between n+1 and 2*n into n pairs (xi,yi) such that the 2*n numbers yi+i and yi-i are all different. - Michel Marcus, Apr 27 2016

			For n = 4, ((1,7), (2,5), (3,8), (4,6)) is an instance of such grouping. ((2,5), (1,7), (3,8), (4,6)) is considered to be the same grouping.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jordan Bell, Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309 (2009), pp 1-31.
M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, p. 240.
G. B. Huff, On pairings of the first 2n natural numbers, Acta. Arith. 23 (1973) 117-126.
D. A. Klarner, The Problem of Reflecting Queens, The American Mathematical Monthly, Vol. 74, No. 8 (Oct., 1967), pp. 953-955.
M. Slater, Number theory Research Problem 1, Bull. Amer. Math. Soc. 69 (1963), 333.

Cf. A000170, A002968, A051223, A051224, A272363.

a(n) = {nb = 0; for (j=0, n!-1, vp = numtoperm(n, j); vb = vector(n, k, vp[k]+n); vs = vector(n, k, vb[k]+k); vd = vector(n, k, vb[k]-k); if (#vs + #vd == #Set(concat(vs, vd)), nb++); ); nb; } \\ Michel Marcus,  Apr 27 2016

a(18)-a(21) from Sean A. Irvine, Jan 13 2018
a(0)-a(3) prepended by Michel Marcus, Oct 03 2018
a(22) from Sean A. Irvine, Oct 04 2018
a(23) from Sean A. Irvine, Oct 07 2018

cp's OEIS Frontend

A051223 Number of ways of placing n nonattacking superqueens on an n X n board.

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A102388 Number of ways of placing n nonattacking Queens of the Night on an n X n board.

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A172200 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n board.

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A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.

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A174642 Number of ways to place 4 nonattacking amazons (superqueens) on a 4 X n board.

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A352661 Number of doubly symmetric characteristic solutions to the n-superqueens problem.

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A352662 Number of singly symmetric characteristic solutions to the n-superqueens problem.

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A352663 Number of asymmetric characteristic solutions to the n-superqueens problem.

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A007631 Number of solutions to non-attacking reflecting queens problem.

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