A051223 -id:A051223 - cp's OEIS frontend

A137774 Number of ways to place n nonattacking empresses on an n X n board.

1, 2, 2, 8, 20, 94, 438, 2766, 19480, 163058, 1546726, 16598282, 197708058, 2586423174, 36769177348, 563504645310, 9248221393974, 161670971937362, 2996936692836754, 58689061747521430, 1210222434323163704, 26204614054454840842, 594313769819021397534, 14086979362268860896282

Offset: 1

Vaclav Kotesovec, Jan 27 2011

An empress moves like a rook and a knight.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Separators - a new statistic for permutations, arXiv:1905.12364 [math.CO], 2019.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, On the Sparseness of the Downsets of Permutations via Their Number of Separators, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 3, Article #S2R21.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.685 and 636.
W. Schubert, N-Queens page

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774, A245011.
Cf. A218244, A002464, A110128, A117574, A089222, A002493.

Asymptotics (Vaclav Kotesovec, Jan 26 2011): a(n)/n! -> 1/e^4.
General asymptotic formulas for number of ways to place n nonattacking pieces rook + leaper[r,s] on an n X n board:
a(n)/n! -> 1/e^2 for 0a(n)/n! -> 1/e^4 for 0

Terms a(16)-a(17) from Vaclav Kotesovec, Feb 06 2011
Terms a(18)-a(19) from Wolfram Schubert, Jul 24 2011
Terms a(20)-a(24) (computed by Wolfram Schubert), Vaclav Kotesovec, Aug 25 2012

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

A051224 Number of ways of placing n nonattacking superqueens on n X n board (symmetric solutions count only once).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 22, 239, 653, 4089, 25411, 166463, 1115871, 8062150, 61984976, 497236090, 4261538564, 38352532487, 360400504834, 3518014210402, 35752764285788

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

A superqueen moves like a queen and a knight.

Superqueens are also called amazons.

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

D. Bill, Durango Bill's The N-Queens Problem
W. Schubert, N-Queens page

Cf. A051223, A002562, A352661, A352662, A352663.

a(n) = (1/8) * (Q(n) + P(n) + 2 * R(n)), where Q(n) = A051223(n) [all solutions], P(n) [point symmetric solutions (180 degrees)] and R(n) [rotationally symmetric solutions (90 degrees)]. This formula has the same structure as the formula for A002562. There seem to be no OEIS sequences (yet) for P(n) and R(n). See the N-Queens page link. - W. Schubert, Nov 29 2009

a(20) from Bill link added Jul 25 2006
a(21)..a(22) added from Bill's website. Max Alekseyev, Oct 19 2008
Added formula and a(23)..a(25) derived by formula. W. Schubert, Nov 29 2009
Added a(26). W. Schubert, Jan 18 2011

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

A225553 Longest checkmate in king and amazon versus king endgame on an n X n chessboard.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

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

			Longest win on an 8x8 chessboard: Ka1 AMb1 - Kd4, 1.AMb1-f5! Kd4-c4! 2.Ka1-b1 Kc4-b4! 3.Kb1-b2 Kb4-a4 4.AMf5-c5#, therefore a(8) = 4.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p.364 (2013), p. 544 (second edition, 2017).

Cf. A172200, A051223, A178967, A225551, A225552, A225554, A225555, A225556, A225557, A227437.

Conjecture: for n > 10, a(n) = floor((n+2)/2).

Empirical g.f.: -x^4*(x^9-x^8+x^4-x-1) / ((x-1)^2*(x+1)). - Colin Barker, May 11 2013

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).

A189864 Number of ways to place n nonattacking composite pieces queen + leaper[1,3] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 56, 176, 1932, 4188, 26960, 182456, 1132064, 7645784, 58695136, 470822912, 3792417988, 32440237692

Offset: 1

Vaclav Kotesovec, Apr 29 2011

(In fairy chess, the leaper [1,3] is called a camel.)

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

Cf. A051223, A189358.

A189865 Number of ways to place n nonattacking composite pieces queen + leaper[1,4] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 2, 10, 0, 0, 4, 32, 76, 196, 632, 3368, 12532, 79788, 468286, 2815088, 18287968, 126620984, 938037664, 7232141830, 59774887344

Offset: 1

Vaclav Kotesovec, Apr 29 2011

In fairy chess the leaper [1,4] is called a giraffe.

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

Cf. A051223, A189864, A189563

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cp's OEIS Frontend

A137774 Number of ways to place n nonattacking empresses 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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A051224 Number of ways of placing n nonattacking superqueens on n X n board (symmetric solutions count only once).

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

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A225553 Longest checkmate in king and amazon versus king endgame on an n X n chessboard.

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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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