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 1-8 of 8 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

Views

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.

A169777 Number of geometrically distinct open knight's tours of a 3 X n chessboard.

Original entry on oeis.org

3, 0, 0, 14, 104, 146, 773, 2698, 14350, 32296, 161714, 460022, 2159794, 5851548, 26468357, 76442996, 330719293, 965759972, 4056479056, 12186078360, 49668414086, 151760518296, 604396415979, 1879966906486, 7330203447133, 23126786408904, 88609897281582
Offset: 4

Views

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

Examples

			The three distinct 3x4 tours were published by Euler in Memoires Acad. Roy. Sci. (Berlin, 1759), 310-337.

References

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

Links

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

a(n) = A169696(n)/4 + A169776(n)/2.

A169771 Number of open knight's tour diagrams of a 3 X n chessboard that have "type F": the endpoints occur in different columns and agree in color with the cells in the nearest corner.

Original entry on oeis.org

2, 0, 0, 52, 224, 520, 1616, 10320, 37024, 125120, 441200, 1798576, 6327472, 22985504, 81178008, 301420176, 1057619944, 3818476576, 13412523392, 48285742208, 168992600680, 602349395456, 2106360581920, 7471875943776, 26073917403304, 92017860990176, 320713651212384
Offset: 4

Views

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

References

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

Links

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

Asymptotic value: 0.02789*3.45059^n.

A169772 Number of open knight's tour diagrams of a 3 X n chessboard that have "type B": the endpoints occur in different columns and disagree in color with the cells in the nearest corner.

Original entry on oeis.org

2, 0, 0, 0, 92, 0, 1064, 0, 14928, 0, 156416, 0, 1785600, 0, 19416704, 0, 211014544, 0, 2261999424, 0, 24067157192, 0, 254242274472, 0, 2669251156032, 0, 27880294589248
Offset: 4

Views

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

References

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

Links

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

A169772(n)=0 unless n mod 2 = 0.
Asymptotic value: 0.00144*n*3.11949^n when n is even.

A169773 Number of open knight's tour diagrams of a 3 X n chessboard that are symmetric under 180-degree rotation and have "type X": both endpoints occur in the same column.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 16, 0, 0, 0, 264, 0, 0, 0, 2144, 0, 0, 0, 22408, 0, 0, 0, 211808, 0, 0, 0, 2087344, 0, 0, 0, 20207664, 0, 0, 0, 197082624, 0, 0, 0, 1916054112, 0, 0, 0, 18652927040, 0, 0, 0, 181485750208, 0, 0, 0, 1766199186560, 0, 0, 0
Offset: 4

Views

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

References

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

Links

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

A169773(n)=0 unless n mod 4 = 1.

Extensions

a(31)-a(60) from Andrew Howroyd, Jul 01 2017

A169775 Number of open knight's tour diagrams of a 3 X n chessboard that are symmetric under 180-degree rotation and have "type B": the endpoints occur in different columns and disagree in color with the cells in the nearest corner.

Original entry on oeis.org

2, 0, 0, 0, 8, 0, 16, 0, 48, 0, 200, 0, 616, 0, 1832, 0, 6008, 0, 19304, 0, 62180, 0, 189580, 0, 615792, 0, 1895952, 0, 6136708, 0, 18699436, 0, 60490008, 0, 184450888, 0, 595959276, 0, 1811054676, 0, 5847417040, 0, 17754996288, 0, 57292227492
Offset: 4

Views

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

References

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

Links

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

A169775(n)=0 unless n mod 2 = 0.

Extensions

a(31)-a(48) from Andrew Howroyd, Jul 01 2017

A169776 Number of geometrically distinct open knight's tours of a 3 X n chessboard that have twofold symmetry.

Original entry on oeis.org

2, 0, 0, 2, 10, 12, 22, 60, 76, 160, 292, 652, 1148, 2600, 3870, 9152, 13710, 32792, 48112, 116624, 171732, 428064, 589842, 1496508, 2069766, 5348640, 7164172, 18742712, 25160796, 66758832, 86664762, 232553036, 302742306, 821495496, 1044549008
Offset: 4

Views

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

References

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

Links

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Formula

A169776(n) = (A169773(n) + A169774(n) + A169775(n))/2.

Extensions

a(31)-a(38) from Andrew Howroyd, Jul 01 2017

A169774 Number of open knight's tour diagrams of a 3 X n chessboard that are symmetric under 180-degree rotation and have "type F": the endpoints occur in different columns and agree in color with the cells in the nearest corner.

Original entry on oeis.org

2, 0, 0, 4, 12, 20, 28, 120, 104, 304, 384, 1304, 1680, 4936, 5908, 18304, 21412, 63440, 76920, 233248, 281284, 833720, 990104, 2993016, 3523740, 10485472, 12432392, 37485424, 44184884, 131430320, 154630088, 465106072, 544994604, 1622783328, 1904647128
Offset: 4

Views

Author

N. J. A. Sloane, May 10 2010, based on a communication from Don Knuth, Apr 28 2010

Keywords

References

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

Links

Crossrefs

Cf. A070030, A169696, A169764-A169777.

Extensions

a(31)-a(38) from Andrew Howroyd, Jul 01 2017
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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