A083386 -id:A083386 - cp's OEIS frontend

A079137 Number of (undirected) Hamiltonian paths on the 4 X n knight graph.

0, 0, 8, 0, 82, 744, 6378, 31088, 189688, 1213112, 6683852, 36486328, 201282470, 1083585304, 5706117458, 29819231288, 154430502724, 790787799376, 4014945695196, 20241304810488, 101336136490228, 504096313001272, 2493533648002492, 12270473056485396

Offset: 1

Eric W. Weisstein, Dec 28 2002

nonn

Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, p. 263, 1942.

Andrew Howroyd, Table of n, a(n) for n = 1..500
George Jellis, Knight's tour diagrams
Colin Rose, Knight's tours
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Knight Graph

See A079312 for 4 times these numbers, A123935 for twice these numbers, A123936 for these numbers halved.

Cf. A169696, A083386, A165134, A328341.

More terms from André Pönitz (poenitz(AT)htwm.de), Jun 11 2003
Edited by N. J. A. Sloane, Oct 30 2006, following suggestions from Colin Rose
Terms a(22) and beyond from Andrew Howroyd, Jul 01 2017

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

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

A308131 Number of (undirected) Hamiltonian paths in the n X n knight graph.

Original entry on oeis.org

0, 0, 0, 0, 864, 3318960, 82787609160, 9795914085489952

Offset: 1

Eric W. Weisstein, May 14 2019

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

Cf. A165134, A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), A306283 (7 X n).

a(n) = A165134(n)/2.

cp's OEIS Frontend

A079137 Number of (undirected) Hamiltonian paths on the 4 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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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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A308131 Number of (undirected) Hamiltonian paths in the n X n knight graph.

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