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.

Showing 1-5 of 5 results.

A169696 Number of undirected Knight's tours on a 3 X n board.

Original entry on oeis.org

0, 0, 0, 8, 0, 0, 52, 396, 560, 3048, 10672, 57248, 128864, 646272, 1838784, 8636880, 23400992, 105865688, 305753680, 1322849752, 3862974304, 16225820000, 48744080192, 198673312880, 607041217056, 2417584484232, 7519864632928, 29320809649000, 92507134938336
Offset: 1

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Author

N. J. A. Sloane, Apr 14 2010, based on a communication from Don Knuth

Keywords

Comments

I think the (old) name "Number of open Knight's tours on a 3 X n board" is somewhat incorrect, because included are those tours in which the start/end cells are knight-neighbors. Such tours are potentially closed, although actually closing them would deprive them of specific start/end cells. "Number of undirected Knight's tours on a 3 X n board" would be a better name. For example the 3x10 has 3048 undirected tours, which would be 6096 directed tours, in accord with Colin Rose results (http://www.tri.org.au/knightframe.html, Solutions:3xm). Note that the 3x10 also has 16 closed tours (A169764 Number of closed Knight's tours on a 3 X n board), and each of those closed tour appears 30 times among the 3048 undirected tours, and 60 times among the 6096 directed tours. - Pierre Charland, Feb 15 2011

References

  • D. E. Knuth, Long and skinny knight's tours, in Selected Papers on Fun and Games, to appear, 2010.

Links

Crossrefs

Cf. A118067.

Formula

a(n) = A169770(n) + A169771(n) + A169772(n).
Asymptotic value: 0.02789*3.45059^n.

A083386 Number (undirected) Hamiltonian paths in the 5 X n knight graph.

Original entry on oeis.org

0, 0, 0, 82, 864, 18784, 622868, 18061054, 264895640, 7886117822, 128411926952, 3611823644006, 56348098488396, 1548284436152798, 24535910156176100, 650456890341276338, 10364916108987670024, 267426031403733462626
Offset: 1

Views

Author

André Pönitz (poenitz(AT)htwm.de), Jun 11 2003

Keywords

Comments

The number of (directed) knight's tours is twice this. - Colin Rose, Jun 12 2006

Examples

			There are 864 knight's tours on a 5 X 5 chessboard.

References

  • Jellis: http://home.freeuk.net/ktn/5a.htm [broken link ?]

Links

Crossrefs

Cf. A079137, A118067.

A165134 Number of directed Hamiltonian paths in the n X n knight graph.

Original entry on oeis.org

1, 0, 0, 0, 1728, 6637920, 165575218320, 19591828170979904
Offset: 1

Views

Author

[No name given] (c.candide(AT)free.fr), Sep 04 2009

Keywords

Comments

Previous name was: Number of knight's paths visiting each square of an n X n chessboard exactly once.

Examples

			From _Gheorghe Coserea_, Oct 08 2016: (Start)
For n=5 the numbers in the table below give the number of knight's paths starting at the respective position on the 5 X 5 chessboard.  In total there are a(5) = 304*4 + 56*8 + 64 = 1728 solutions.
    [1]  [2]  [3]  [4]  [5]
[1] 304    0   56    0  304
[2]   0   56    0   56    0
[3]  56    0   64    0   56
[4]   0   56    0   56    0
[5] 304    0   56    0  304
(End)

Links

Crossrefs

Cf. Undirected Hamiltonian paths: A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), A306283 (7 X n), A308131 (n X n).
Cf. A001230, A118067, A306282.

Extensions

a(7) from Guenter Stertenbrink, added by Alex Chernov, Sep 01 2013
a(1)=1, a(2)=0 prepended by Max Alekseyev, Sep 22 2013
a(8) from Alex Chernov, May 10 2014
Name made more precise by Eric W. Weisstein, Apr 14 2019

A306282 Triangle read by rows: T(n,k) is the number of (directed) Hamiltonian paths in the n X k knight graph.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 164, 1728, 0, 0, 0, 1488, 37568, 6637920, 0, 0, 104, 12756, 1245736, 779938932, 165575218320, 0, 0, 792, 62176, 36122108
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2019

Keywords

Examples

			Triangle begins:
n\k | 1  2    3      4        5          6             7
----+----------------------------------------------------
1   | 1;
2   | 0, 0;
3   | 0, 0,   0;
4   | 0, 0,  16,     0;
5   | 0, 0,   0,   164,    1728;
6   | 0, 0,   0,  1488,   37568,   6637920;
7   | 0, 0, 104, 12756, 1245736, 779938932, 165575218320;

Links

Crossrefs

T(n,n) gives A165134.
Cf. A118067, A123935.

A347363 Number of self-avoiding knight's paths from the lower left corner to the lower right corner of a 3 X n chessboard.

Original entry on oeis.org

1, 0, 2, 8, 32, 156, 871, 5292, 28702, 154162, 845532, 4662014, 25579463, 140098348, 767973001, 4212065280, 23097682805, 126643657272, 694390484065, 3807499106946, 20877386149018, 114474503105178, 627683328355315, 3441701959286326, 18871492466212538
Offset: 1

Views

Author

Andrzej Kukla, Aug 29 2021

Keywords

Comments

If we enumerate the squares in the 3 X n board like this:
------------------------------------
| 1 | 4 | 7 | 10 | 13 | ... | 3n-2 |
------------------------------------
| 2 | 5 | 8 | 11 | 14 | ... | 3n-1 |
------------------------------------
| 3 | 6 | 9 | 12 | 15 | ... | 3n |
------------------------------------
then a(n) is the number of self-avoiding knight's paths on such a board from square 3 to square 3n.

Examples

			For n = 4 we have exactly 8 self-avoiding paths starting at square 3 and ending at square 12:
  3,  4,  9, 10,  5, 12;
  3,  4,  9,  2,  7, 12;
  3,  8,  1,  6,  7, 12;
  3,  4, 11,  6,  7, 12;
  3,  8,  1,  6, 11,  4,  9,  2,  7, 12;
  3,  4, 11,  6,  7,  2,  9, 10,  5, 12;
  3,  8,  1,  6,  7,  2,  9, 10,  5, 12;
  3,  8,  1,  6, 11,  4,  9, 10,  5, 12;

Crossrefs

Cf. A118067, A169696, A212715.

Extensions

a(8)-a(15) from Pontus von Brömssen, Aug 30 2021
Terms a(16) and beyond from Andrew Howroyd, Nov 19 2021
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