A000170 -id:A000170 - cp's OEIS frontend

A225623 Number of ways to arrange 2n queens on an n X n chessboard, with no more than 2 queens in each row, column or diagonal.

0, 1, 2, 11, 92, 1097, 19448, 477136, 14244856, 537809179, 24194010708, 1317062528249

Offset: 1

Vaclav Kotesovec, Aug 04 2013

This problem is slightly different from A000769 or A219760. In the first example on an 8 x 8 board, the queens c7, d5 and e3 (or queens a2, c5 and e8) are in a line, but such case is allowed. The elementary step can be only [0,1], [1,0] or [1,1], not for example [1,2] or [2,3].

Vaclav Kotesovec, Examples

Cf. A000170, A000769, A001499, A001501, A058528, A075754, A172544, A172541.

Definition clarified by Vaclav Kotesovec, Dec 18 2014

a(10)-a(12) from Martin Ehrenstein, Jan 09 2022

A225740 Number of arrangements of n nonattacking Queens on an n X n board, with no Queens on both main diagonals.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 12, 8, 12, 64, 760, 2080, 11524, 77672, 454376, 2972720, 19274208, 160500016, 1150812148, 9797154696, 79646705916, 771589104392

Offset: 1

Vaclav Kotesovec, Aug 04 2013

			From _Martin Ehrenstein_, Jan 10 2022: (Start)
Solutions for n=9 filtered from the Coserea link in A000170:
  3 5 8 2 9 7 1 4 6
  3 6 9 2 8 1 4 7 5
  4 6 9 3 1 8 2 5 7
  5 3 6 9 2 8 1 4 7
  5 7 4 1 8 2 9 6 3
  6 4 1 7 9 2 8 5 3
  7 4 1 8 2 9 6 3 5
  7 5 2 8 1 3 9 6 4
(End)

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.447
Vaclav Kotesovec, example, board 8x8, no queens on line a1-h8 and no queens on line a8-h1

Cf. A000170, A003471.

a(9)-a(19) corrected and extended with a(20)-a(23) by Martin Ehrenstein, Jan 10 2022

a(9)-a(19) confirmed by Vaclav Kotesovec, Jan 11 2022

A352325 Number of ways of placing A352241(n) nonattacking black-square queens on an n X n board.

Original entry on oeis.org

1, 2, 5, 6, 8, 6, 10, 112, 104, 80, 40, 36, 2172, 1414, 984, 384, 240, 70396, 39400, 22468, 3696, 1152, 4457616, 2246138, 1060976, 185932

Offset: 1

Vaclav Kotesovec, Mar 12 2022

Cf. A000170, A352241.

a(20)-a(26) from Martin Ehrenstein, Mar 12 2022

A359441 The n-Queens Constant.

Original entry on oeis.org

1, 9, 4, 4, 0, 0

Offset: 1

Vaclav Kotesovec, Jan 01 2023

Lower bound: 1.944000752019729...

Upper bound: 1.9440010813092217...

			1.94400...

Don Knuth, Xqueens and Xqueenons, 2021.
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.
Parth Nobel, Akshay Agrawal and Stephen Boyd, Computing tighter bounds on the n-queens constant via Newton’s method, Sep 17 2022.
Michael Simkin, The number of n-queens configurations, arXiv:2107.13460 [math.CO], 2021.
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. A000170.

Lim_{n->infinity} A000170(n)^(1/n)/n = exp(-A359441).

A375103 Number of ways of placing the maximum number of nonattacking queens on a 3-dimensional chessboard of order n.

Original entry on oeis.org

1, 8, 16, 1344, 1056, 912, 96, 24

Offset: 1

Tim Kunt, Jul 30 2024

Tim Kunt, The n-Queens Problem in Higher Dimensions, arXiv:2406.06260 [math.OC], 2024.
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set.

Cf. A000170, A068940, A375104.

a(8) from Tim Kunt, Aug 12 2024

A375104 Number of ways of placing the maximum number of nonattacking queens on a 4-dimensional chessboard of order n.

Original entry on oeis.org

1, 16, 4992, 404564

Offset: 1

Tim Kunt, Jul 30 2024

Tim Kunt, The n-Queens Problem in Higher Dimensions.
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set.

Cf. A000170, A068941, A375103.

A375800 Number of ways of placing 2n nonattacking rooks on a hexagonal board of equilateral triangular spaces with n spaces along each edge.

Original entry on oeis.org

3, 24, 348, 7812, 255756, 11747504, 714121392

Offset: 1

Hugh Robinson, Aug 29 2024

A rook move on the equilateral triangle tessellation is a move along a path through successively edge-adjacent faces, that turns alternately left and right at each face (starting with either left or right), so that the overall direction of movement remains approximately parallel to one set of triangle edges.

Also counts the number of 3 X 2n matrices such that each row is a permutation of {1, .., 2n}, the first row is the identity permutation (1 .. 2n), and each column sums to either 3n+1 or 3n+2. This parallels how the equivalent problem for the board tessellated with hexagons (A002047) counts the number of 3 X (2n-1) zero-sum arrays.

			For n = 2, the a(2) = 24 arrangements are rotations and reflections of:
      o---o---o           o---o---o           o---o---o
     /X\ / \ / \         /X\ / \ / \         /X\ / \ / \
    o---o---o---o       o---o---o---o       o---o---o---o
   / \ / \ /X\ / \     / \ / \ / \X/ \     / \ / \ / \X/ \
  o---o---o---o---o   o---o---o---o---o   o---o---o---o---o
   \ / \ / \ / \X/     \ / \ / \ /X\ /     \ /X\ / \ / \ /
    o---o---o---o       o---o---o---o       o---o---o---o
     \X/ \ / \ /         \X/ \ / \ /         \ / \ / \X/
      o---o---o           o---o---o           o---o---o
   (12 symmetries)      (6 symmetries)      (6 symmetries)
For n = 2, the a(2) = 24 matrices counted are:
 1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
 2  3  4  1     2  3  4  1     2  4  1  3     2  4  3  1
 4  2  1  3     4  3  1  2     4  2  3  1     4  1  2  3
-
 1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
 2  4  3  1     3  1  4  2     3  2  4  1     3  2  4  1
 4  2  1  3     3  4  1  2     3  4  1  2     4  3  1  2
-
 1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
 3  4  1  2     3  4  1  2     3  4  2  1     3  4  2  1
 4  1  3  2     4  2  3  1     4  1  2  3     4  1  3  2
plus the same matrices with rows 2 and 3 interchanged.

Cf. A002047, A000170, A000142.

% minizinc -D 'N=5' -s --all-solutions a375800.mzn
include "globals.mzn";
include "alldifferent.mzn";
int: N;
array[1..2*N] of var 1..2*N: perm1;
array[1..2*N] of var 1..2*N: perm2;
constraint forall(i in 1..2*N)(3*N+1 <= perm1[i]+perm2[i]+i /\ perm1[i]+perm2[i]+i <= 3*N+2);
constraint alldifferent(perm1);
constraint alldifferent(perm2);
solve satisfy;
output [show(i) ++ " " | i in 1..2*N];
output [show(perm1[i]) ++ " " | i in 1..2*N];
output [show(perm2[i]) ++ " " | i in 1..2*N];

A059963 Triangle T(n,k) giving number of ways of placing n nonattacking queens on n X n board with the queen on the first row fixed at column k, 1<=k<=n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 2, 0, 1, 1, 1, 1, 0, 4, 7, 6, 6, 6, 7, 4, 4, 8, 16, 18, 18, 16, 8, 4, 28, 30, 47, 44, 54, 44, 47, 30, 28, 64, 48, 65, 93, 92, 92, 93, 65, 48, 64, 96, 219, 209, 295, 346, 350, 346, 295, 209, 219, 96, 500, 806, 1165, 1359, 1631, 1639

Offset: 1

Yong Kong (ykong(AT)curagen.com), Mar 03 2001

A000170 (non-attacking queens) can be derived from this sequence as follows: a(12)= 2*(S1(12)+S2(12)+S3(12)+S4(12)+S5(12)+S6(12)) when n is even, a(13)=S7(13) + 2*(S1(13)+S2(13)+S3(13)+S4(13)+S5(13)+S6(13)) when n is odd. Here Si(j) means T(j,i). - Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), Jan 05 2004

			When n = 8 there are 16 ways to place if the queen on the first row is at the third column
Triangle begins:
1,
0,0,
0,0,0,
0,1,1,0,
2,2,2,2,2,
0,1,1,1,1,0,
4,7,6,6,6,7,4,
4,8,16,18,18,16,8,4,
28,30,47,44,54,44,47,30,28, etc.

Patrick R. GUILLEMIN, Extension of triangle to 22 rows
Patrick R. GUILLEMIN, Extension of triangle to 22 rows

Cf. A000170.

Confirmed by Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), who, together with colleagues, has computed the first 21 rows of this triangle, Jan 05 2004

Sep 15 2004: Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), together with colleagues, has computed the 22nd row of this triangle.

A062165 Number of ways of placing n nonattacking (normal) queens on n X n board, solutions similar on the torus count only once.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 4, 13, 36, 115, 813, 3083, 21001, 131859, 868613

Offset: 1

Matthias Engelhardt

Two n-queens solutions p and q are considered similar iff there is a factor f, 0 < f < n, satisfying gcd (f,n) = 1, such that for all k from {0, ..., n-1} q (k * f mod n) = p (k) * f mod n or q is a rotation, a reflection or a shift of such a q. In other words, also expansions are allowed which move the queen at (k, p(k)) to (f * k mod n, f * p(k) mod n).

The sequence reduces exactly the objects of A062164 and, via that sequence, these of A002562 and A000170. Note that the equivalence classes of this sequence are a subset of A062168.

M. Engelhardt, The N queens problem

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

A129551 Number of ways to place n+2 queens and 2 pawns on an n X n board so that no two queens attack each other.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 44, 280, 1304, 12452, 105012, 977664, 9239816, 90776620, 897446092

Offset: 1

R. Douglas Chatham (d.chatham(AT)moreheadstate.edu), Apr 20 2007

			a(4)=0 because when 6 queens are placed on a 4 X 4 board, at least two queens will be adjacent and therefore mutually attacking.

R. D. Chatham, The N+k Queens Problem Page.
R. D. Chatham, M. Doyle, G. H. Fricke, J. Reitmann, R. D. Skaggs and M. Wolff, Independence and Domination Separation in Chessboard Graphs, J. Combin. Math. Combin. Comput. 68 (2009), 3-17.

Cf. A000170, A129552.

cp's OEIS Frontend

A225623 Number of ways to arrange 2n queens on an n X n chessboard, with no more than 2 queens in each row, column or diagonal.

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A225740 Number of arrangements of n nonattacking Queens on an n X n board, with no Queens on both main diagonals.

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A352325 Number of ways of placing A352241(n) nonattacking black-square queens on an n X n board.

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A359441 The n-Queens Constant.

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A375103 Number of ways of placing the maximum number of nonattacking queens on a 3-dimensional chessboard of order n.

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A375104 Number of ways of placing the maximum number of nonattacking queens on a 4-dimensional chessboard of order n.

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A375800 Number of ways of placing 2n nonattacking rooks on a hexagonal board of equilateral triangular spaces with n spaces along each edge.

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MiniZinc

A059963 Triangle T(n,k) giving number of ways of placing n nonattacking queens on n X n board with the queen on the first row fixed at column k, 1<=k<=n.

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Extensions

A062165 Number of ways of placing n nonattacking (normal) queens on n X n board, solutions similar on the torus count only once.

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Extensions

A129551 Number of ways to place n+2 queens and 2 pawns on an n X n board so that no two queens attack each other.

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