cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A189866 Number of ways to place n nonattacking composite pieces queen + leaper[1,5] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 2, 10, 0, 0, 0, 40, 52, 160, 500, 2656, 11540, 67776, 415716, 2610520, 17450592, 124903880, 944965832, 7292031780
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A051223, A189864, A189865, A189564

A352662 Number of singly symmetric characteristic solutions to the n-superqueens problem.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 11, 49, 79, 245, 498, 1192, 3798, 11594
Offset: 1

Views

Author

Don Knuth, Mar 25 2022

Keywords

Comments

Superqueens are also called amazons. They combine the moves of queen and knight.

Examples

			For n=10 the a(10)=1 solution is
  +---------------------+
  | . . . A . . . . . . |
  | . . . . . . . A . . |
  | A . . . . . . . . . |
  | . . . . A . . . . . |
  | . . . . . . . . A . |
  | . A . . . . . . . . |
  | . . . . . A . . . . |
  | . . . . . . . . . A |
  | . . A . . . . . . . |
  | . . . . . . A . . . |
  +---------------------+

References

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

Crossrefs

Cf. A051223, A051224, A352661, A352663.

A352663 Number of asymmetric characteristic solutions to the n-superqueens problem.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 18, 231, 642, 4040, 25320, 166201, 1115373, 8060958, 61981118, 497224414
Offset: 1

Views

Author

Don Knuth, Mar 25 2022

Keywords

Examples

			One of the a(11)=5 solutions is
  +-----------------------+
  | A . . . . . . . . . . |
  | . . . . A . . . . . . |
  | . . . . . . . . A . . |
  | . A . . . . . . . . . |
  | . . . . . A . . . . . |
  | . . . . . . . . . A . |
  | . . A . . . . . . . . |
  | . . . . . . A . . . . |
  | . . . . . . . . . . A |
  | . . . A . . . . . . . |
  | . . . . . . . A . . . |
  +-----------------------+
and the other four are obtained by wraparound shifts.

References

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

Crossrefs

Cf. A051223, A051224, A352661, A352662.

A007631 Number of solutions to non-attacking reflecting queens problem.

Original entry on oeis.org

1, 1, 0, 0, 2, 4, 0, 2, 10, 32, 38, 140, 496, 1186, 3178, 16792, 82038, 289566, 1139874, 5914118, 33800010, 142337180, 721286448, 4384569864
Offset: 0

Views

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

Comments

a(n) is the number of ways to pair the natural numbers from 1 to n with those between n+1 and 2*n into n pairs (xi,yi) such that the 2*n numbers yi+i and yi-i are all different. - Michel Marcus, Apr 27 2016

Examples

			For n = 4, ((1,7), (2,5), (3,8), (4,6)) is an instance of such grouping. ((2,5), (1,7), (3,8), (4,6)) is considered to be the same grouping.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000170, A002968, A051223, A051224, A272363.

Programs

  • PARI
    a(n) = {nb = 0; for (j=0, n!-1, vp = numtoperm(n, j); vb = vector(n, k, vp[k]+n); vs = vector(n, k, vb[k]+k); vd = vector(n, k, vb[k]-k); if (#vs + #vd == #Set(concat(vs, vd)), nb++); ); nb; } \\ Michel Marcus,  Apr 27 2016

Extensions

a(18)-a(21) from Sean A. Irvine, Jan 13 2018
a(0)-a(3) prepended by Michel Marcus, Oct 03 2018
a(22) from Sean A. Irvine, Oct 04 2018
a(23) from Sean A. Irvine, Oct 07 2018

A189867 Number of ways to place n nonattacking composite pieces queen + leaper[2,3] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 2, 10, 0, 0, 0, 0, 48, 152, 472, 2696, 12320, 74436, 429620, 2515116, 16122496, 113016608, 843492920, 6575649316, 54694203188
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

In fairy chess the leaper [2,3] is called a zebra.
a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)|<>3 AND |p(j+3)-p(j)|<>2 AND |p(m+k)-p(m)|<>k for all i>=1, j>=1, m>=1, k>=1, i+2<=n, j+3<=n, m+k<=n

Links

Crossrefs

Cf. A051223, A189864, A189565

A189868 Number of ways to place n nonattacking composite pieces queen + leaper[2,4] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 0, 0, 4, 64, 384, 2936, 10720, 62664, 420272, 2540584, 16373760, 115200080, 868564232, 6708291360
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A051223, A189865, A189867, A189566

A189869 Number of ways to place n nonattacking composite pieces queen + leaper[2,5] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 2, 10, 4, 28, 20, 56, 72, 288, 1304, 6368, 22884, 125864, 755412, 4565572, 29862256, 207238124, 1540506028, 11774180220
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A051223, A189866, A189867, A189868, A189567

A133143 Maximal number of mutually nonattacking Super Queens on an n X n board. (A Super Queen is a queen with both queen and knight powers.)

Original entry on oeis.org

1, 1, 1, 2, 4, 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Offset: 1

Views

Author

Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Dec 16 2007

Keywords

Comments

From Vaclav Kotesovec, Mar 14 2011: (Start)
For n >= 10, a(n)=n, see A051223.
For same problem on a toroidal chessboard the results for n > 10 are the same as for queens (A085801). (End)

Examples

			a(4) = 2:
|X|0|0|0|
|0|0|0|X|
|0|0|0|0|
|0|0|0|0|
where X denotes the place of a Super Queen.

Links

Programs

  • Java
    // See Java code text file in the links section.

A189870 Number of ways to place n nonattacking composite pieces queen + leaper[3,4] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 2, 2, 4, 28, 0, 20, 52, 280, 1192, 5520, 20196, 115936, 701836, 4174032, 27261284, 193428616, 1445733328, 11133210948
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A051223, A189865, A189868, A189568

A189871 Number of ways to place n nonattacking composite pieces queen + leaper[3,5] on an n X n chessboard.

Original entry on oeis.org

1, 0, 0, 2, 10, 0, 8, 24, 72, 116, 384, 1660, 7344, 24364, 130408, 743360, 4242704, 27018788, 190618152, 1431986780, 11053915716
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+3)-p(i)|<>5 AND |p(j+5)-p(j)|<>3 AND |p(m+k)-p(m)|<>k for all i>=1, j>=1, m>=1, k>=1, i+3<=n, j+5<=n, m+k<=n

Links

Crossrefs

Cf. A051223, A189866, A189869, A189569, A189870
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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