A070030 -id:A070030 - cp's OEIS frontend

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.

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

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

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 4..1000
George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
D. E. Knuth Generating functions for A169770-A169777 and A169696.

Cf. A070030, A169696, A169764-A169777.

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

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

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 4..1000
George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
D. E. Knuth Generating functions for A169770-A169777 and A169696.

Cf. A070030, A169696, A169764-A169777.

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

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

nonn

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

George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
D. E. Knuth Generating functions for A169770-A169777 and A169696.

Cf. A070030, A169696, A169764-A169777.

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

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

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

nonn

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

George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
D. E. Knuth Generating functions for A169770-A169777 and A169696.

Cf. A070030, A169696, A169764-A169777.

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

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

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

nonn

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

George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
D. E. Knuth Generating functions for A169770-A169777 and A169696.

Cf. A070030, A169696, A169764-A169777.

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

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

A175855 The number of closed Knight's tours on a 5 X 2n board.

Original entry on oeis.org

0, 0, 8, 44202, 13311268, 4557702762, 1495135512514, 491857035772330, 161514101568508400, 53034853662012222798, 17414154188157170439208, 5717847862749642677204182, 1877435447920358266870897874, 616447390029326136628439042672, 202407848349722353779265745190616, 66459727085467788423206394162537418, 21821760546806761707309514948565417796, 7165079447164571822068029945303172129766, 2352622444655438705806553391345493395131580, 772473271844923268504474277422663237674924998

Offset: 1

Johan de Ruiter, Dec 05 2010

nonn

			The smallest 5 X 2n board admitting a closed Knight's tour is the 5 X 6, on which there are 8 such tours.

J. de Ruiter, Counting Domino Coverings and Chessboard Cycles, 2010.

A070030 deals with 3 X 2n boards, A175881 deals with 6 X n boards.

Cf. A383661.

a(n) = A383661(5n). - Don Knuth, May 05 2025

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

A169769 Number of geometrically distinct closed knight's tours of a 3 X n chessboard.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 0, 44, 0, 396, 0, 3868, 0, 37070, 0, 362192, 0, 3516314, 0, 34237842, 0, 333077332, 0, 3241403380, 0, 31542464952, 0, 306944118820, 0, 2986962829456, 0, 29066627247828, 0, 282854730020224, 0, 2752516325518516, 0

Offset: 4

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

nonn

			The six solutions for n=10 were first published by Kraitchik in 1927.

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

Seiichi Manyama, Table of n, a(n) for n = 4..2031 (terms 4..1000 from Alois P. Heinz)
George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).

Cf. A070030, A169696, A169764-A169777.

a(n) = A169764(n)/4 + A169768(n)/2.
a(n) = 0 unless n mod 2 = 0.
Generating function: 2*z^10*((-2*(1 + 5*z^2 - 34*z^4 - 116*z^6 + 505*z^8 + 616*z^10 - 3179*z^12 - 4*z^14 + 9536*z^16 - 8176*z^18 - 13392*z^20 + 15360*z^22 + 13888*z^24 + 2784*z^26 - 3328*z^28 - 22016*z^30 + 5120*z^32 + 2048*z^34))/
(-1 + 6*z^2 + 64*z^4 - 200*z^6 - 1000*z^8 + 3016*z^10 + 3488*z^12 - 24256*z^14 + 23776*z^16 + 104168*z^18 - 203408*z^20 - 184704*z^22 + 443392*z^24 + 14336*z^26 - 151296*z^28 + 145920*z^30 - 263424*z^32 + 317440*z^34 + 36864*z^36 - 966656*z^38 + 573440*z^40 + 131072*z^42) -
(1 + 6*z^6 - 31*z^8 + 8*z^10 + 53*z^12 - 179*z^14 + 312*z^16 - 84*z^18 - 1280*z^20 + 1974*z^22 - 1232*z^24 - 858*z^26 + 10320*z^28 - 8154*z^30 + 5556*z^32 + 9972*z^34 - 35152*z^36 + 11992*z^38 - 37920*z^40 - 35856*z^42 + 47488*z^44 - 3888*z^46 + 103264*z^48 + 45344*z^50 - 12608*z^52 + 19520*z^54 - 30336*z^56 + 11072*z^58 - 35328*z^60 - 28160*z^62 - 84480*z^64 - 56832*z^66 + 12288*z^68 + 24576*z^70 + 40960*z^72 + 8192*z^74 + 16384*z^76)/
(-1 + 6*z^4 + 64*z^8 - 200*z^12 - 1000*z^16 + 3016*z^20 + 3488*z^24 - 24256*z^28 + 23776*z^32 + 104168*z^36 - 203408*z^40 - 184704*z^44 + 443392*z^48 + 14336*z^52 - 151296*z^56 + 145920*z^60 - 263424*z^64 + 317440*z^68 + 36864*z^72 - 966656*z^76 + 573440*z^80 + 131072*z^84)).

More terms from R. J. Mathar, Oct 09 2010

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

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

nonn

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

George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
D. E. Knuth Generating functions for A169770-A169777 and A169696.

Cf. A070030, A169696, A169764-A169777.

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

cp's OEIS Frontend

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.

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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.

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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.

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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.

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

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A175855 The number of closed Knight's tours on a 5 X 2n board.

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

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A169769 Number of geometrically distinct closed knight's tours of a 3 X n chessboard.

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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.

Views

Author

Keywords

References

Links

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