A178721 -id:A178721 - cp's OEIS frontend

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

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

A036464 Number of ways to place two nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 8, 44, 140, 340, 700, 1288, 2184, 3480, 5280, 7700, 10868, 14924, 20020, 26320, 34000, 43248, 54264, 67260, 82460, 100100, 120428, 143704, 170200, 200200, 234000, 271908, 314244, 361340, 413540, 471200, 534688, 604384

Offset: 1

Robert G. Wilson v, Raymond Bush (c17h21no4(AT)hotmail.com), Kirk Conely, N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, January 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.
V. Kotesovec, Non-attacking chess pieces
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A047659, A061994, A108792, A176186, A178721.

Column k=2 of A348129.

f:=n->n^4/2 - 5*n^3/3 + 3*n^2/2 - n/3; [seq(f(n),n=1..200)]; # N. J. A. Sloane, Feb 16 2013

f[k_] := 2 k; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]   (* A036464 *)
Table[a[n]/4, {n, 2, 50}] (* A000914 *)
(* Clark Kimberling, Dec 31 2011 *)
CoefficientList[Series[4 x^2 (2 + x) / (1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,8,44,140},50] (* Harvey P. Dale, Mar 26 2015 *)

a(n) = C(n, 3)*(3*n-1).
G.f.: 4*x^3*(2+x)/(1-x)^5. - Colin Barker, May 02 2012
a(n) = 2*sum_{i=1..n-2} i(i + 1)^2. - Wesley Ivan Hurt, Mar 18 2014
E.g.f.: (exp(x) * x^3 * (8 + 3*x))/6. - Vaclav Kotesovec, Feb 15 2015
For n>0, a(n) = A163102(n-1) - A006331(n-1). - Antal Pinter, Sep 20 2015

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.

A180402 a(n) = lcm(1,...,Fibonacci(n)).

Original entry on oeis.org

1, 1, 2, 6, 60, 840, 360360, 232792560, 144403552893600, 164249358725037825439200, 718766754945489455304472257065075294400, 33312720618553145840562713089120360606823375590405920630576000

Offset: 1

Vaclav Kotesovec, Sep 02 2010

nonn

Also least period for number of ways of placing k non-attacking queens on an n X n chessboard. [conjectured by Kotesovec; proved for n <= 5. - Thomas Zaslavsky, Jun 24 2018]

Alois P. Heinz, Table of n, a(n) for n = 1..17
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. See Table 8.1.
V. Kotesovec, Non-attacking chess pieces, 6ed, p.31, 2013

Cf. A178717, A178721, A078491, A218492, A035105, A062954.

a:= n-> ilcm($1..(<<0|1>, <1|1>>^n)[1,2]):
seq(a(n), n=1..14);  # Alois P. Heinz, Aug 12 2017

Table[Apply[LCM, Range[Fibonacci[k]]], {k, 1, 10}]
Array[LCM @@ Range@Fibonacci@# &, 12] (* Robert G. Wilson v, Sep 05 2010 *)

a(n) = lcm([1..fibonacci(n)]); \\ Michel Marcus, Jun 24 2018

a(11) onwards from Robert G. Wilson v, Sep 05 2010

A252593 Number of ways to place 8 nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 92, 13848, 636524, 14803480, 207667564, 2008758532, 14752426528, 87154016752, 432539436508, 1858901487620

Offset: 1

Antal Pinter, Dec 18 2014

Conjectured recurrence order is 477 (see "Non-attacking chess pieces", p. 19). - Vaclav Kotesovec, Dec 19 2014

S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-Queens Problem, I. General theory, arXiv:1303.1879 [math.CO], 2013-2014.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
Antal Pinter, Combinatorics, software for enumerating positions of non-attacking chess pieces.
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.

Cf. A000170, A036464, A047659, A061994, A108792, A176186, A178721.

Column k=8 of A348129.

a(n) = n^16/40320 - n^15/432 + 221*n^14/2160 + O(n^13). - Vaclav Kotesovec, Dec 19 2014

a(16) from Vaclav Kotesovec, Dec 19 2014

a(17) from Vaclav Kotesovec, Dec 20 2014

cp's OEIS Frontend

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

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A036464 Number of ways to place two nonattacking queens on an n X n board.

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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.

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A180402 a(n) = lcm(1,...,Fibonacci(n)).

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A252593 Number of ways to place 8 nonattacking queens on an n X n board.

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A238844 Combinatorial configuration types of n (unlabeled) queens on a square board.

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