A079137 -id:A079137 - cp's OEIS frontend

A123936 a(n) = A079137(n)/2.

0, 0, 4, 0, 41, 372, 3189, 15544, 94844, 606556, 3341926, 18243164, 100641235, 541792652, 2853058729, 14909615644, 77215251362, 395393899688, 2007472847598, 10120652405244, 50668068245114, 252048156500636, 1246766824001246, 6135236528242698, 30045694433419662

Offset: 1

N. J. A. Sloane, Oct 30 2006

nonn

Number of nonequivalent (or geometrically distinct) directed open knight's tours on a 4 X n chessboard up to rotation and reflection. See A328341 for the number of geometrically distinct undirected open knights's tours. - Andrew Howroyd, Oct 12 2019

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A079137, A123935, A328341.

Terms a(22) and beyond from Andrew Howroyd, Oct 13 2019

A079312 a(n) = 4 * A079137(n).

Original entry on oeis.org

0, 0, 32, 0, 328, 2976, 25512, 124352, 758752, 4852448, 26735408, 145945312, 805129880, 4334341216, 22824469832, 119276925152, 617722010896, 3163151197504, 16059782780784, 80965219241952, 405344545960912

Offset: 1

Alexander D. Healy, Feb 11 2003

nonn

See A079137, which is the main entry for this problem.

Equals 4*A079137(n). Cf. A070030.

Incorrect description removed by Andrew Howroyd, Oct 12 2019

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

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

nonn

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

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

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

Andrew Howroyd, Table of n, a(n) for n = 1..50
Hochschule Mittweida, Graph Editor
C. Rose, The Distribution of the Knight
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Knight Graph

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

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

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

			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)

Stefan Behnel, The Knight's Paths
A. Chernov, Open knight's tours
Gheorghe Coserea, Solutions for 5x5 chessboard
P. Hingston, G. Kendall, Enumerating knight's tours using an ant colony algorithm, The 2005 IEEE Congress on Evolutionary Computation, 2 (2006), 1003-1010
G. Stertenbrink, Number of Knight's Tours
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Knight Graph

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.

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

A118067 Number of (directed) Hamiltonian paths in the 3 X n knight graph.

Original entry on oeis.org

0, 0, 0, 16, 0, 0, 104, 792, 1120, 6096, 21344, 114496, 257728, 1292544, 3677568, 17273760, 46801984, 211731376, 611507360, 2645699504, 7725948608, 32451640000, 97488160384, 397346625760, 1214082434112, 4835168968464, 15039729265856, 58641619298000

Offset: 1

Colin Rose, May 11 2006

nonn

1. Jelliss computes the number of tour diagrams (which is equal to half the number of tours). 2. Sequence A079137 computes the number of tour DIAGRAMS for a 4 X k board (again, equal to half the number of tours). 3. Kraitchik (1942) incorrectly reports 376 tour diagrams for the 3 X 8 case; the correct number is 396 (i.e., 792 tours) [cf. Rose, Jelliss].

Kraitchik, M., Mathematical Recreations. New York: W. W. Norton, pp. 264-5, 1942.

Seiichi Manyama, Table of n, a(n) for n = 1..1861
G. Jelliss, Open Knight's Tours of Three-Rank Boards
Seiichi Manyama calculated a(14)-a(21) by yoh2's code
C. Rose, The Distribution of the Knight.
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Knight Graph

Cf. A169696, A079137, A083386, A165134.

Cf. A158074. - Eric W. Weisstein, Mar 13 2009

Mathematica notebook available at: http://www.tri.org.au/knightframe.html

a(n) = 2 * A169696(n). - Andrew Howroyd, Jul 01 2017

a(13) from Eric W. Weisstein, Mar 13 2009
a(14)-a(21) from Seiichi Manyama, Apr 25 2016
a(22)-a(28) from Andrew Howroyd, Jul 01 2017

A123935 Number of (directed) Hamiltonian paths on the 4 X n knight graph.

Original entry on oeis.org

0, 0, 16, 0, 164, 1488, 12756, 62176, 379376, 2426224, 13367704, 72972656, 402564940, 2167170608, 11412234916, 59638462576, 308861005448, 1581575598752, 8029891390392, 40482609620976, 202672272980456, 1008192626002544, 4987067296004984, 24540946112970792

Offset: 1

N. J. A. Sloane, Oct 30 2006

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Knight Graph

Cf. A079137, A306282.

a(n) = 2*A079137(n).

Terms a(22) and beyond from Andrew Howroyd, Oct 13 2019

A306281 Number of (undirected) Hamiltonian paths on the 6 X n knight graph.

Original entry on oeis.org

0, 0, 0, 744, 18784, 3318960, 389969466, 24964893804, 1770631206422, 143827657320448, 10668015492137018, 763955912402146956, 55382275594728895388, 4008456113318585117624, 285329658478008271167456, 20203324505809248032547768, 1425847547641332606081597198, 100103728161914529291271962728

Offset: 1

Seiichi Manyama, Feb 03 2019

nonn

Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Knight Graph

Cf. A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), this sequence (6 X n), A306283 (7 X n), A308131 (n X n).

a(8)-a(11) from Andrew Howroyd, Oct 14 2019
a(12) from Valentin Gubarev, Dec 23 2024
a(13)-a(18) from Andrew Howroyd, Dec 26 2024

A306283 Number of (undirected) Hamiltonian paths on the 7 X n knight graph.

Original entry on oeis.org

0, 0, 52, 6378, 622868, 389969466, 82787609160, 20666425060328, 2903212163753000, 1025241126020698238

Offset: 1

Seiichi Manyama, Feb 03 2019

Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Knight Graph.

Cf. A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), this sequence (7 X n).

a(8)-a(10) from Valentin Gubarev, Dec 23 2024

A309273 Number of semi-magic (only short lines are magic) knight's tours on a 4 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 9, 16, 38, 104, 267, 608, 1444, 3480, 8221, 19212, 45262, 213280, 250247, 587072, 1378912, 3237456

Offset: 1

Awani Kumar, Jul 20 2019

			Example 4 X 7 semi-magic knight's tour (only short lines are magic):
  +----+----+----+----+----+----+----+
  |  9 | 28 |  7 | 18 |  3 | 24 | 13 |
  +----+----+----+----+----+----+----+
  |  6 | 17 | 10 | 25 | 14 | 21 |  2 |
  +----+----+----+----+----+----+----+
  | 27 |  8 | 15 |  4 | 19 | 12 | 23 |
  +----+----+----+----+----+----+----+
  | 16 |  5 | 26 | 11 | 22 |  1 | 20 |
  +----+----+----+----+----+----+----+
.
Example 4 X 16 semi-magic knight's tour (only short lines are magic):
   1 62  3 36 29 38 19 58 25 44 17 56 23 52 15 54
  32 35 30 61  4 59 26 45 18 57 24 43 16 55 12 51
  63  2 33 28 37  6 39 20 47  8 41 22 49 10 53 14
  34 31 64  5 60 27 46  7 40 21 48  9 42 13 50 11

G. P. Jelliss, Semi-magic Knight's Tours
Awani Kumar, Studies in Tours of Knight on Rectangular Boards, arXiv:1802.09340 [math.GM], 2018.

Cf. A079137, A123936, A309271.

A328341 Number of geometrically distinct open knight's tours on a 4 X n chessboard.

Original entry on oeis.org

0, 0, 3, 0, 22, 186, 1603, 7772, 47478, 303278, 1671273, 9121582, 50322028, 270896326, 1426536267, 7454807822, 38607660199, 197696949844, 1003736587788, 5060326202622, 25334034892953, 126024078250318, 623383415637750, 3067618264121349, 15022847233751804, 73245459228339114

Offset: 1

Andrew Howroyd, Oct 12 2019

nonn

			a(3) = 3 because there are two symmetric and one asymmetric tour:
  +----+----+----+----+   +----+----+----+----+   +----+----+----+----+
  |  8 | 11 |  6 |  3 |   |  1 |  4 |  7 | 10 |   |  1 |  4 |  7 | 10 |
  +----+----+----+----+   +----+----+----+----+   +----+----+----+----+
  |  1 |  4 |  9 | 12 |   |  8 | 11 |  2 |  5 |   | 12 |  9 |  2 |  5 |
  +----+----+----+----+   +----+----+----+----+   +----+----+----+----+
  | 10 |  7 |  2 |  5 |   |  3 |  6 |  9 | 12 |   |  3 |  6 | 11 |  8 |
  +----+----+----+----+   +----+----+----+----+   +----+----+----+----+

George Jellis, Knight's tours of Four Rank Boards

Cf. A079137, A123936, A169777, A328340.

a(2*n) = A123936(2*n)/2; a(2*n-1) = (A123936(2*n-1) + A328340(n))/2.

cp's OEIS Frontend

A123936 a(n) = A079137(n)/2.

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A079312 a(n) = 4 * A079137(n).

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

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

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A118067 Number of (directed) Hamiltonian paths in the 3 X n knight graph.

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A123935 Number of (directed) Hamiltonian paths on the 4 X n knight graph.

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A306281 Number of (undirected) Hamiltonian paths on the 6 X n knight graph.

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A306283 Number of (undirected) Hamiltonian paths on the 7 X n knight graph.

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A309273 Number of semi-magic (only short lines are magic) knight's tours on a 4 X n board.

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A328341 Number of geometrically distinct open knight's tours on a 4 X n chessboard.

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