A079137
Number of (undirected) Hamiltonian paths on the 4 X n knight graph.
Original entry on oeis.org
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
- Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, p. 263, 1942.
See
A079312 for 4 times these numbers,
A123935 for twice these numbers,
A123936 for these numbers halved.
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
A001230
Number of undirected closed knight's tours on a 2n X 2n chessboard.
Original entry on oeis.org
0, 0, 9862, 13267364410532
Offset: 1
- Brendan McKay, personal communication, Feb 03, 1997.
- W. W. Rouse Ball, Mathematical Recreations and Essays (various editions), Chap. 6.
- I. Wegener, Branching Programs and Binary Decision Diagrams, SIAM, Philadelphia, 2000; see p. 369.
- G. L. Chia, Siew-Hui Ong, Generalized knight's tour on rectangular chessboards, Disc. Appl. Math. 150(1-3) (2005) 80-98.
- N. D. Elkies and R. P. Stanley, The mathematical knight, Math. Intelligencer, 25 (No. 1) (2003), 22-34.
- Brady Haran, Knight's Tour, Numberphile video (2014).
- George Jelliss, Knight's Tour Notes
- Stoyan Kapralov, Valentin Bakoev, and Kaloyan Kapralov, Enumeration of Some Closed Knight Paths, arXiv preprint arXiv:1711.06792 [math.CO], 2017.
- M. Loebbing and I. Wegener, The Number of Knight's Tours Equals 33,439,123,484,294 -- Counting with Binary Decision Diagrams. Electronic Journal of Combinatorics 3 (1996), R5. [The number given in the paper is incorrect, see comments.]
- B. D. McKay, "Knight's Tours of an 8x8 Chessboard". Technical Report TR-CS-97-03, Department of Computer Science, Australian National University (1997). [Cached copy, with permission]
- Eric Weisstein's World of Mathematics, Hamiltonian Cycle
- Eric Weisstein's World of Mathematics, Knight Graph
- Wikipedia, Knight's tour
-
Table[Length[FindHamiltonianCycle[KnightTourGraph[2 n, 2 n], All]], {n, 3}]
Loebbing and Wegener incorrectly gave 33439123484294 for the 8 X 8 board. The value given here is due to
Brendan McKay and agrees with that given by Wegener in his book.
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
- Kraitchik, M., Mathematical Recreations. New York: W. W. Norton, pp. 264-5, 1942.
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
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;
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
A289204
Number of (undirected) paths in the n X n knight graph.
Original entry on oeis.org
0, 0, 56, 14980, 19005336, 278982789260
Offset: 1
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