A000170 -id:A000170 - cp's OEIS frontend

A181501 Triangle read by rows: number of solutions of n queens problem for given n and given number of connection components of conflict constellation.

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 28, 0, 4, 8, 0, 0, 0, 0, 0, 0, 92, 0, 0, 0, 0, 0, 0, 0, 8, 272, 56, 16, 0, 0, 0, 0, 0, 0, 96, 344, 240, 44, 0, 0, 0, 0, 0, 0

Offset: 0

Matthias Engelhardt, Oct 30 2010

The rightmost part of the triangle contains only zeros. As any connection component needs at least two queens, the number of connection components of a solution is always less than or equal to n.

			Triangle begins:
   0;
   1, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 2, 0, 0;
  10, 0, 0, 0, 0, 0;
   0, 4, 0, 0, 0, 0, 0;
  28, 0, 4, 8, 0, 0, 0, 0;
  ... - _Andrew Howroyd_, Dec 31 2017
for n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two connection components in the conflicts graph. So, the terms for n=4 are 0, 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

M. Engelhardt, Rows n=0..16 of triangle, flattened
Matthias Engelhardt, Conflicts in the n-queens problem
Matthias Engelhardt, Conflict tables for the n-queens problem
M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.

Cf. A000170, A007705, A181499, A181500, A181502.

Row sum =A000170 (number of n queens placements)

Column 0 has same values as A007705 (torus n queens solutions)

Offset corrected by Andrew Howroyd, Dec 31 2017

A181502 Triangle read by rows: number of solutions of n queens problem for given n and given maximal size of a connection component in the conflict constellation.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 28, 8, 4, 0, 0, 0, 0, 0, 0, 64, 24, 4, 0, 0, 0, 0, 0, 0, 248, 80, 16, 8, 0, 0, 0, 0, 0, 0, 172, 484, 36, 32, 0, 0, 0

Offset: 0

Matthias Engelhardt, Oct 30 2010

Torus solutions, i.e. solutions having an empty conflict constellation, are counted in column 1; this is caused by an interpretation of a queen not engaged in any conflict as an island in the conflict graph. Using the definition strictly, these queens should be removed from the graph and the numbers should appear in column 0, not column 1.

			Triangle begins:
  0;
  0,  1;
  0,  0, 0;
  0,  0, 0, 0;
  0,  0, 2, 0, 0;
  0, 10, 0, 0, 0, 0;
  0,  0, 0, 0, 4, 0, 0;
  0, 28, 8, 4, 0, 0, 0, 0;
... - _Andrew Howroyd_, Dec 31 2017
for n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two conflicts So the terms for n=4 are 0 (0 solutions for n=4 having 0 conflicts), 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

M. Engelhardt, Rows n=0..16 of triangle, flattened
Matthias Engelhardt, Conflicts in the n-queens problem
Matthias Engelhardt, Conflict tables for the n-queens problem
M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.

Cf. A000170, A007705, A181499, A181500, A181501.

Row sum =A000170 (number of n queens placements)
Column 1 has same values as A007705 (torus n queens solutions)
Column 0 is always zero.

Offset corrected by Andrew Howroyd, Dec 31 2017

A189839 Number of ways to place n nonattacking composite pieces rook + rider[3,3] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 20, 80, 384, 2112, 12992, 94272, 716800, 6141440, 58451568, 596647568, 6555879072, 77766001056, 981202169600

Offset: 1

Vaclav Kotesovec, Apr 29 2011

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

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

Cf. A117574, A000170, A189838

A189852 Number of ways to place n nonattacking composite pieces rook + rider[1,5] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 336, 1474, 8340, 57756, 475658, 2812910, 20852460, 181255892, 1817101242, 20435345782, 197871434994

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+k)-p(i)|<>5k AND |p(j+5k)-p(j)|<>k for all i>=1, j>=1, k>=1, i+k<=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, A189837, A189850, A189851

A189853 Number of ways to place n nonattacking composite pieces rook + rider[1,6] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 2640, 13596, 87768, 680274, 6090756, 61678252, 482005340, 4454053680, 46705656280, 549750105234

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+k)-p(i)|<>6k AND |p(j+6k)-p(j)|<>k for all i>=1, j>=1, k>=1, i+k<=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, A189837, A189850, A189851, A189852

A189854 Number of ways to place n nonattacking composite pieces rook + rider[2,3] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 12, 36, 174, 500, 2052, 12112, 65092, 407882, 2954798, 20568796, 157579774, 1346294112, 11580692142, 110130002110, 1145065547108

Offset: 1

Vaclav Kotesovec, Apr 29 2011

(in fairy chess the rider [2,3] is called a Zebrarider)

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+2k)-p(i)|<>3k AND |p(j+3k)-p(j)|<>2k for all i>=1, j>=1, k>=1, i+2k<=n, j+3k<=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, A189837, A189850

A189855 Number of ways to place n nonattacking composite pieces rook + rider[2,4] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 60, 208, 1184, 7840, 36036, 209664, 1395480, 10996728, 83573220, 723835856, 7132494776, 77976981216, 790552134804

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+2k)-p(i)|<>4k AND |p(j+4k)-p(j)|<>2k for all i>=1, j>=1, k>=1, i+2k<=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, A189854, A189837

A189856 Number of ways to place n nonattacking composite pieces rook + rider[2,5] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 392, 1810, 10400, 72228, 589674, 3823906, 29420944, 266232984, 2711139976, 30669073348, 316482938974

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+2k)-p(i)|<>5k AND |p(j+5k)-p(j)|<>2k for all i>=1, j>=1, k>=1, i+2k<=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, A189854, A189855

A134593 a(n) = 5n^2 + 10n + 1. Coefficients of the rational part of (1 + sqrt(n))^5.

Original entry on oeis.org

1, 16, 41, 76, 121, 176, 241, 316, 401, 496, 601, 716, 841, 976, 1121, 1276, 1441, 1616, 1801, 1996, 2201, 2416, 2641, 2876, 3121, 3376, 3641, 3916, 4201, 4496, 4801, 5116, 5441, 5776, 6121, 6476, 6841, 7216, 7601, 7996, 8401, 8816, 9241, 9676, 10121

Offset: 0

Artur Jasinski, Nov 04 2007

(1+sqrt(n))^5 = (5*n^2 + 10*n + 1) + (n^2 + 10*n + 5)*sqrt(n). Coefficients of the irrational part are A134594.

Number of entries required to describe the options and constraints in Don Knuth's formulation of the n nonattacking queens on an n X n board problem (A000170) as input for his DLX (Dancing Links eXact coverage) program. Can be seen as "entries successfully read" in the video from his 2018 Annual Christmas Lecture. - Hugo Pfoertner, Jan 09 2019

D. E. Knuth, Donald Knuth's 24th Annual Christmas Lecture: Dancing Links, Stanfordonline, Video published on YouTube, Dec 12, 2018.
Takao Komatsu, Ritika Goel, and Neha Gupta, The Frobenius number for the triple of the 2-step star numbers, arXiv:2409.14788 [math.CO], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000170, A134594.

Table[(5n^2 + 10n + 1), {n, 0, 50}]
LinearRecurrence[{3,-3,1},{1,16,41},50] (* Harvey P. Dale, Oct 20 2023 *)

a(n)=5*n^2+10*n+1 \\ Charles R Greathouse IV, Jun 17 2017

print([5*i**2-4 for i in range(1,100)])
# Ruskin Harding, Mar 27 2013

G.f.: (4*x^2 - 13*x - 1)/(x-1)^3. - R. J. Mathar, Nov 14 2007
a(n) = a(n-1) + 10*n + 5 (with a(0)=1). - Vincenzo Librandi, Nov 23 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, May 17 2021
E.g.f.: exp(x)*(1 + 15*x + 5*x^2). - Stefano Spezia, Sep 27 2024

Edited by Charles R Greathouse IV, Aug 09 2010

A189840 Number of ways to place n nonattacking composite pieces rook + rider[4,4] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 108, 544, 3264, 23040, 171072, 1409664, 12916224, 131217408, 1428028032, 16709309440, 210367491840, 2847184825728

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(j+4k)-p(j)|<>4k for all j>=1, k>=1, 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. A189255, A000170, A189838, A189839

cp's OEIS Frontend

A181501 Triangle read by rows: number of solutions of n queens problem for given n and given number of connection components of conflict constellation.

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A181502 Triangle read by rows: number of solutions of n queens problem for given n and given maximal size of a connection component in the conflict constellation.

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A189839 Number of ways to place n nonattacking composite pieces rook + rider[3,3] on an n X n chessboard.

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A189852 Number of ways to place n nonattacking composite pieces rook + rider[1,5] on an n X n chessboard.

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A189853 Number of ways to place n nonattacking composite pieces rook + rider[1,6] on an n X n chessboard.

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A189854 Number of ways to place n nonattacking composite pieces rook + rider[2,3] on an n X n chessboard.

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A189855 Number of ways to place n nonattacking composite pieces rook + rider[2,4] on an n X n chessboard.

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A189856 Number of ways to place n nonattacking composite pieces rook + rider[2,5] on an n X n chessboard.

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A134593 a(n) = 5*n^2 + 10*n + 1. Coefficients of the rational part of (1 + sqrt(n))^5.

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A189840 Number of ways to place n nonattacking composite pieces rook + rider[4,4] on an n X n chessboard.

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A134593 a(n) = 5n^2 + 10n + 1. Coefficients of the rational part of (1 + sqrt(n))^5.