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.

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A306283 Number of (undirected) Hamiltonian paths on the 7 X n knight graph.

Original entry on oeis.org

0, 0, 52, 6378, 622868, 389969466, 82787609160, 20666425060328, 2903212163753000, 1025241126020698238
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2019

Keywords

Links

Crossrefs

Cf. A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), this sequence (7 X n).

Extensions

a(8)-a(10) from Valentin Gubarev, Dec 23 2024

A169766 Number of closed knight's tour diagrams of a 3 X n chessboard that have "Bergholtian symmetry".

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 48, 0, 152, 0, 352, 0, 1200, 0, 3752, 0, 12912, 0, 34768, 0, 122120, 0, 346128, 0, 1202240, 0, 3337424, 0, 11650864, 0, 32634960, 0, 113539392, 0, 316870592, 0, 1104442752, 0, 3086894528, 0, 10748713792, 0, 30023935744, 0
Offset: 4

Views

Author

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

Keywords

Comments

When the board is rotated 180 degrees, the diagram remains the same, but the tour reverses direction.

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) = 0 unless n mod 2 = 0.
Generating function: (2*(2*z^10 - 8*z^14 + 24*z^16 - 76*z^18 + 32*z^20 + 288*z^22 - 716*z^24 + 792*z^26 - 336*z^28 - 2908*z^30 + 7896*z^32 - 1464*z^34 - 3432*z^36 + 7416*z^38 - 32616*z^40 - 11792*z^42 + 39888*z^44 + 35472*z^46 + 47968*z^48 + 35776*z^50 - 143424*z^52 - 197824*z^54 - 15552*z^56 - 11008*z^58 + 181376*z^60 + 269440*z^62 + 78080*z^64 + 53760*z^66 + 44288*z^68 - 48128*z^70 - 112640*z^72 - 124928*z^74 - 227328*z^76 - 155648*z^78 + 98304*z^80 + 147456*z^82 + 32768*z^84))/
(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)

Extensions

More terms extracted from the g.f. by R. J. Mathar, Oct 09 2010

A169768 Number of geometrically distinct closed knight's tours of a 3 X n chessboard that have twofold symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 24, 0, 24, 0, 276, 0, 176, 0, 2604, 0, 1876, 0, 25736, 0, 17384, 0, 248816, 0, 173064, 0, 2424608, 0, 1668712, 0, 23581056, 0, 16317480, 0, 229513584, 0, 158435296, 0, 2233386048, 0, 1543447264, 0, 21733496960
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

a(n) = (A169765(n)+A169766(n)+A169767(n))/2.
a(n) = 0 unless n mod 2 = 0.
Generating function: (4*z^10 + 24*z^16 - 124*z^18 + 32*z^20 + 212*z^22 - 716*z^24 + 1248*z^26 - 336*z^28 - 5120*z^30 + 7896*z^32 - 4928*z^34 - 3432*z^36 + 41280*z^38 - 32616*z^40 + 22224*z^42 + 39888*z^44 - 140608*z^46 + 47968*z^48 - 151680*z^50 - 143424*z^52 + 189952*z^54 - 15552*z^56 + 413056*z^58 + 181376*z^60 - 50432*z^62 + 78080*z^64 - 121344*z^66 + 44288*z^68 - 141312*z^70 - 112640*z^72 - 337920*z^74 - 227328*z^76 + 49152*z^78 + 98304*z^80 + 163840*z^82 + 32768*z^84 + 65536*z^86)/
(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).

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

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

Views

Author

Eric W. Weisstein, May 14 2019

Keywords

Links

Crossrefs

Cf. A165134, A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), A306283 (7 X n).

Formula

a(n) = A165134(n)/2.

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

Views

Author

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

Keywords

Examples

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

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) = 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)).

Extensions

More terms from R. J. Mathar, Oct 09 2010
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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