A169764 -id:A169764 - cp's OEIS frontend

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.

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

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

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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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

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References

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