A000170 -id:A000170 - cp's OEIS frontend

A129553 Number of ways to place n+3 queens and 3 pawns on an n X n board so that no two queens attack each other.

0, 0, 0, 0, 0, 0, 0, 8, 44, 528, 5976, 77896, 1052884, 13666360

Offset: 1

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

			a(4)=0 because when 7 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, A129554.

A171760 The maximum number of sets of n queens which can be placed on an n X n chessboard such that no queen attacks another queen in the same set.

Original entry on oeis.org

0, 1, 0, 0, 2, 5, 4, 7, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17

Offset: 0

Howard A. Landman, Dec 17 2009

a(n) is nonzero for n >= 4 (there is always at least one solution to the n-queens problem). a(n) <= n (because n sets of n queens fill up the board). a(n) = n if n = 1 or 5 (mod 6).
a(n) is at least two for all even n >= 4 since a solution and its reflection will fit on the same board. - Charlie Neder, Jul 24 2018
In addition a(18) >= 16 and a(20) = 20. - Benjamin Butin, Dec 11 2023

			a(4) = 2 because there are only two solutions to the 4-queens problem and they can both fit on the same board:
 0 1 2 0
 2 0 0 1
 1 0 0 2
 0 2 1 0
a(8) = 6 since at least 6 solutions to the 8-queens problem can fit on the same board but 7 solutions can't:
 3 0 5 2 1 6 0 4
 0 1 4 0 5 3 2 6
 4 6 0 1 2 0 5 3
 5 2 3 6 0 4 1 0
 6 4 1 5 0 2 3 0
 2 5 0 3 4 0 6 1
 0 3 2 0 6 1 4 5
 1 0 6 4 3 5 0 2
a(9) = 7
 7 5 6 3 1 . . 2 4
 6 3 . 4 2 7 1 . 5
 . . 2 7 5 6 3 4 1
 4 7 5 1 . 2 . 6 3
 3 1 4 . 6 . 7 5 2
 . 6 . 5 3 4 2 1 7
 2 4 7 6 . 1 5 3 .
 5 . 1 2 7 3 4 . 6
 1 2 3 . 4 5 6 7 .
a(10) = 8
 3 4 2 8 . . 1 7 5 6
 6 . 7 1 5 4 8 2 . 3
 . 1 5 6 7 2 3 4 8 .
 2 8 4 . 3 6 . 5 1 7
 7 . 6 5 1 8 4 3 . 2
 8 3 . 4 2 7 5 . 6 1
 5 6 8 7 . . 2 1 3 4
 4 7 3 . 8 1 . 6 2 5
 . 5 1 2 6 3 7 8 4 .
 1 2 . 3 4 5 6 . 7 8

Benjamin Butin, Solution for a(14) = 14
Giovanni Resta, A C program for computing a(1)-a(11)

Cf. A000170.

a(6) and known a(7) added by Charlie Neder, Jul 24 2018
a(8)-a(10) and known a(11)-a(13) from Giovanni Resta, Jul 26 2018
a(14) from Benjamin Butin, Nov 07 2023
a(15)-a(17) from Benjamin Butin, Dec 11 2023

A189842 Number of ways to place n nonattacking composite pieces rook + rider[6,6] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 4800, 34752, 280512, 2528256, 25282560, 278323200, 3242649600, 40330371072, 536528954880, 7633092132864

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(j+6k)-p(j)|<>6k for all j>=1, k>=1, j+6k<=n

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

Cf. A189271, A000170, A189838, A189839, A189840, A189841

A189858 Number of ways to place n nonattacking composite pieces rook + rider[3,4] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 80, 326, 1566, 9544, 53696, 347382, 2566892, 21907934, 184868860, 1704360992, 17294597926, 192725663600, 2139133978996

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+3k)-p(i)|<>4k AND |p(j+4k)-p(j)|<>3k for all i>=1, j>=1, k>=1, i+3k<=n, j+4k<=n.

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

Cf. A000170, A189851, A189855.

A189861 Number of ways to place n nonattacking composite pieces rook + rider[4,5] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 552, 2826, 17080, 117816, 943250, 7369128, 63533572, 603300392, 6280101222, 71927971040, 836503868762

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+4k)-p(i)|<>5k AND |p(j+5k)-p(j)|<>4k for all i>=1, j>=1, k>=1, i+4k<=n, j+5k<=n

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

Cf. A000170, A189852, A189856, A189859

A189862 Number of ways to place n nonattacking composite pieces rook + rider[4,6] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 3720, 22336, 153796, 1213344, 10849504, 108891704, 1023690268, 10593791168, 119694887008, 1472935989952

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+4k)-p(i)|<>6k AND |p(j+6k)-p(j)|<>4k for all i>=1, j>=1, k>=1, i+4k<=n, j+6k<=n

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

Cf. A000170, A189853, A189857, A189860

A245011 Number of ways to place n nonattacking princesses on an n X n board.

Original entry on oeis.org

1, 4, 6, 86, 854, 9556, 146168, 2660326, 56083228, 1349544632, 36786865968, 1117327217782

Offset: 1

Vaclav Kotesovec, Sep 16 2014

A princess moves like a bishop and a knight.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 744.
Wikipedia, Fairy chess piece

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774.
Cf. A225555.

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

A260189 a(n) = A033148(n) / 2^floor(n/4).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 4, 8, 0, 0, 15, 22, 0, 0, 52, 51, 0, 0, 257, 342, 0, 0, 1589, 2609, 0, 0, 11417, 16896, 0, 0, 75375, 99114, 0, 0, 616010, 876579, 0, 0, 5253278, 8551800, 0, 0, 49667373, 79595269, 0, 0, 525731268, 764804085, 0, 0, 5932910966, 8905825760, 0, 0

Offset: 1

Vaclav Kotesovec, following a suggestion of Don Knuth, Jul 18 2015

W. Ahrens, Mathematische Unterhaltungen und Spiele, 2nd edition, volume 1, Teubner, 1910, pages 249-258.
Maurice Kraitchik, Le probleme des reines, Bruxelles: L'Échiquier, 1926, 18.

P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]

Cf. A002562, A032522, A033148, A037223, A037224.

A269133 Number of ways to place m nonattacking queens on an m X n board, 1 <= m <= n (triangular array).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 4, 2, 5, 12, 14, 12, 10, 6, 20, 36, 46, 40, 4, 7, 30, 76, 140, 164, 94, 40, 8, 42, 140, 344, 568, 550, 312, 92, 9, 56, 234, 732, 1614, 2292, 2038, 1066, 352, 10, 72, 364, 1400, 3916, 7552, 9632, 7828, 4040, 724, 11, 90, 536, 2468, 8492, 21362, 37248, 44148, 34774, 15116, 2680, 12, 110, 756, 4080, 16852, 52856, 120104, 195270, 222720, 160964, 68264, 14200

Offset: 1

Marko Riedel, Feb 19 2016

			The triangular array begins:
   n\m  1   2   3    4     5     6      7      8      9     10    11    12
   1    1
   2    2   0
   3    3   2   0
   4    4   6   4    2
   5    5  12  14   12    10
   6    6  20  36   46    40     4
   7    7  30  76  140   164    94     40
   8    8  42 140  344   568   550    312     92
   9    9  56 234  732  1614  2292   2038   1066    352
  10   10  72 364 1400  3916  7552   9632   7828   4040    724
  11   11  90 536 2468  8492 21362  37248  44148  34774  15116  2680
  12   12 110 756 4080 16852 52856 120104 195270 222720 160964 68264 14200
...

Math StackExchange, State space for eight queen problem
Marko Riedel, Perl program to compute triangular array of nonattacking queens configurations

Cf. A000170 (m=n), A002562, A065256, A348129.
Cf. A000027 (m=1), A002378 (m=2), A061989 (m=3), A061990 (m=4), A061991 (m=5), A061992 (m=6), A061993 (m=7), A172449 (m=8).
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A006717, A051906, A319284 (backtrack trees).

{A269133(m, n, B=[], t=if(#B, setminus(n, Set(concat(B+t=[-#B..-1], B-t))), n=[1..n]))= if(#B < m-1, vecsum([A269133(m, setminus(n, [t]), concat(B,t)) | t<-t]), #t)} \\ M. F. Hasler, Jan 11 2022

cp's OEIS Frontend

A129553 Number of ways to place n+3 queens and 3 pawns on an n X n board so that no two queens attack each other.

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A171760 The maximum number of sets of n queens which can be placed on an n X n chessboard such that no queen attacks another queen in the same set.

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A189842 Number of ways to place n nonattacking composite pieces rook + rider[6,6] on an n X n chessboard.

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A189858 Number of ways to place n nonattacking composite pieces rook + rider[3,4] on an n X n chessboard.

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A189861 Number of ways to place n nonattacking composite pieces rook + rider[4,5] on an n X n chessboard.

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A189862 Number of ways to place n nonattacking composite pieces rook + rider[4,6] on an n X n chessboard.

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A245011 Number of ways to place n nonattacking princesses on an n X n board.

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

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A260189 a(n) = A033148(n) / 2^floor(n/4).

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A269133 Number of ways to place m nonattacking queens on an m X n board, 1 <= m <= n (triangular array).

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