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
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
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
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
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 |
+----+----+----+----+ +----+----+----+----+ +----+----+----+----+
A309271
Number of magic knight's tours on a 4 X 2n board.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 16, 88, 464, 2076, 9904, 47456
Offset: 1
An example of a magic knight's tour on a 4 X 18 board. All rows sum to 657 and all columns sum to 146:
1 70 33 40 5 42 9 66 29 62 27 58 13 52 25 56 19 50
36 39 4 69 32 65 6 43 10 45 14 61 26 57 20 51 24 55
71 2 37 34 41 8 67 30 63 28 59 12 47 16 53 22 49 18
38 35 72 3 68 31 64 7 44 11 46 15 60 21 48 17 54 23
- M. Kraitchik, Mathematical Recreations, Dover, 1953, 257-266.
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