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 1-10 of 23 results. Next

A070030 Number of closed Knight's tours on a 3 X 2n board.

Original entry on oeis.org

0, 0, 0, 0, 16, 176, 1536, 15424, 147728, 1448416, 14060048, 136947616, 1332257856, 12965578752, 126169362176, 1227776129152, 11947846468608, 116266505653888, 1131418872918784, 11010065269439104, 107141489725900544, 1042616896632882688, 10145938076107491328
Offset: 1

Views

Author

Noam D. Elkies, Apr 13 2002

Keywords

Comments

Leonhard Euler stated that no such tours are possible [Memoires Acad. Roy. Sci. (Berlin, 1759), 310-337], and many authors echoed this statement until Ernest Bergholt exhibited solutions for 2n=10 and 12 in British Chess Magazine 1918, page 74. The full set of 16 solutions for 2n=10 was published by Kraitchik in 1927. - Don Knuth, Apr 28 2010
Obtained independently by Don Knuth and Noam D. Elkies in April 1994, both using the transfer-matrix method (in slightly different ways). For this method, see for instance chapter 4.7 of R. P. Stanley's Enumerative Combinatorics, Vol. 1 (1986).
Ian Stewart, Mathematical Recreations column, Scientific American, February 1998, "Feedback", page 95, reports that Richard Ulmer of Denver has sent him a letter reporting work on this subject, about which he is writing a thesis, giving the terms though a(21). - N. J. A. Sloane, Mar 01 2018

Examples

			The smallest 3 X 2n board admitting a closed Knight's tour is the 3 X 10, on which there are 16 such tours.

References

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

Links

Crossrefs

See also A169764, which gives the number of closed Knight's tours on a 3 X n board. Cf. A169696, A169765-A169777, A001230.
Cf. A158074, A383660.

Programs

  • Mathematica
    Rest[CoefficientList[Series[16 * (x^5 + 5*x^6 - 34*x^7 - 116*x^8 + 505*x^9 + 616*x^10 - 3179*x^11 - 4*x^12 + 9536*x^13 - 8176*x^14 - 13392*x^15 + 15360*x^16 + 13888*x^17 + 2784*x^18 - 3328*x^19 - 22016*x^20 + 5120*x^21 + 2048*x^22) / (1 - 6*x - 64*x^2 + 200*x^3 + 1000*x^4 - 3016*x^5 - 3488*x^6 + 24256*x^7 - 23776*x^8 - 104168*x^9 + 203408*x^10 + 184704*x^11 - 443392*x^12 - 14336*x^13 + 151296*x^14 - 145920*x^15 + 263424*x^16 - 317440*x^17 - 36864*x^18 + 966656*x^19 - 573440*x^20 - 131072*x^21), {x, 0, 30}], x]] (* Vaclav Kotesovec, Mar 03 2016 *)
  • PARI
    g = 16 * (z^5 + 5*z^6 - 34*z^7 - 116*z^8 + 505*z^9 + 616*z^10 - 3179*z^11 - 4*z^ 12 + 9536*z^13 - 8176*z^14 - 13392*z^15 + 15360*z^16 + 13888*z^17 + 2784*z^18 - 3328*z^19 - 22016*z^20 + 5120*z^21 + 2048*z^22) / (1 - 6*z - 64*z^2 + 200*z^3 + 1000*z^4 - 3016*z^5 - 3488*z^6 + 24256*z^7 - 23776*z^8 - 104168*z^9 + 203408*z^1 0 + 184704*z^11 - 443392*z^12 - 14336*z^13 + 151296*z^14 - 145920*z^15 + 263424* z^16 - 317440*z^17 - 36864*z^18 + 966656*z^19 - 573440*z^20 - 131072*z^21); g = g + O(z^31); vector(30,n,polcoeff(g,n))

Formula

Generating function: 16 * (z^5 + 5*z^6 - 34*z^7 - 116*z^8 + 505*z^9 + 616*z^10 - 3179*z^11 - 4*z^12 + 9536*z^13 - 8176*z^14 - 13392*z^15 + 15360*z^16 + 13888*z^17 + 2784*z^18 - 3328*z^19 - 22016*z^20 + 5120*z^21 + 2048*z^22) / (1 - 6*z - 64*z^2 + 200*z^3 + 1000*z^4 - 3016*z^5 - 3488*z^6 + 24256*z^7 - 23776*z^8 - 104168*z^9 + 203408*z^10 + 184704*z^11 - 443392*z^12 - 14336*z^13 + 151296*z^14 - 145920*z^15 + 263424*z^16 - 317440*z^17 - 36864*z^18 + 966656*z^19 - 573440*z^20 - 131072*z^21).
Asymptotic value .0001899*3.11949^(2n). - Don Knuth, Apr 28 2010. More decimal places: a(n) ~ 0.0001898644879979384968655648993009439045986223511152141689341... * 3.1194904567551021585814810124470909514449088706168023079811958...^(2*n). - Vaclav Kotesovec, Mar 03 2016
a(n) = A158074(n)/2. - Eric W. Weisstein, Mar 18 2009
Recurrence (D. E. Knuth and N. D. Elkies, 1994): a(n) = 6*a(n-1) + 64*a(n-2) - 200*a(n-3) - 1000*a(n-4) + 3016*a(n-5) + 3488*a(n-6) - 24256*a(n-7) + 23776*a(n-8) + 104168*a(n-9) - 203408*a(n-10) - 184704*a(n-11) + 443392*a(n-12) + 14336*a(n-13) - 151296*a(n-14) + 145920*a(n-15) - 263424*a(n-16) + 317440*a(n-17) + 36864*a(n-18) - 966656*a(n-19) + 573440*a(n-20) + 131072*a(n-21), for n>=23. - Vaclav Kotesovec, Mar 03 2016
a(n) = A383660(3n). - Don Knuth, May 05 2025

Extensions

Comment corrected by Johannes W. Meijer, Nov 22 2010

A169764 Number of closed Knight's tours on a 3 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 176, 0, 1536, 0, 15424, 0, 147728, 0, 1448416, 0, 14060048, 0, 136947616, 0, 1332257856, 0, 12965578752, 0, 126169362176, 0, 1227776129152, 0, 11947846468608, 0, 116266505653888, 0, 1131418872918784, 0, 11010065269439104, 0, 107141489725900544
Offset: 1

Views

Author

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

Keywords

Comments

a(2n) = A070030(n), a(2n+1) = 0.
A070030 is the main entry for this sequence. See that entry for much more information.

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, A169765-A169777.

Programs

  • Mathematica
    CoefficientList[Series[(16*z^10 +80*z^12 -544*z^14 -1856*z^16 +8080*z^18 +9856*z^20 -50864*z^22 -64*z^24 +152576*z^26 -130816*z^28 -214272*z^30 +245760*z^32 +222208*z^34 +44544*z^36 -53248*z^38 -352256*z^40 +81920*z^42 +32768*z^44) / (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), {z,0,50}], z] (* Harvey P. Dale, Feb 12 2013 *)

Formula

Asymptotic value .0001899*3.11949^n when n is even.
Generating function: (16*z^10 + 80*z^12 - 544*z^14 - 1856*z^16 + 8080*z^18 + 9856*z^20 - 50864*z^22 - 64*z^24 + 152576*z^26 - 130816*z^28 - 214272*z^30 + 245760*z^32 + 222208*z^34 + 44544*z^36 - 53248*z^38 - 352256*z^40 + 81920*z^42 + 32768*z^44)/(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).

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.

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

Views

Author

Eric W. Weisstein, Dec 28 2002

Keywords

References

  • Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, p. 263, 1942.

Links

Crossrefs

See A079312 for 4 times these numbers, A123935 for twice these numbers, A123936 for these numbers halved.
Cf. A169696, A083386, A165134, A328341.

Extensions

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
Terms a(22) and beyond from Andrew Howroyd, Jul 01 2017

A169770 Number of open knight's tour diagrams of a 3 X n chessboard that have "type X": both endpoints occur in the same column.

Original entry on oeis.org

4, 0, 0, 0, 80, 40, 368, 352, 5296, 3744, 48656, 40208, 523808, 415488, 5270976, 4333504, 54215264, 44497728, 551297184, 458337984, 5613555008, 4691821600, 56981627840, 47988689152, 577641089664, 489273948160, 5845628996352
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 0.000169*n*3.11949^n when n is even, 0.0000526*n*3.11949^n when n is odd.

A165134 Number of directed Hamiltonian paths in the n X n knight graph.

Original entry on oeis.org

1, 0, 0, 0, 1728, 6637920, 165575218320, 19591828170979904
Offset: 1

Views

Author

[No name given] (c.candide(AT)free.fr), Sep 04 2009

Keywords

Comments

Previous name was: Number of knight's paths visiting each square of an n X n chessboard exactly once.

Examples

			From _Gheorghe Coserea_, Oct 08 2016: (Start)
For n=5 the numbers in the table below give the number of knight's paths starting at the respective position on the 5 X 5 chessboard.  In total there are a(5) = 304*4 + 56*8 + 64 = 1728 solutions.
    [1]  [2]  [3]  [4]  [5]
[1] 304    0   56    0  304
[2]   0   56    0   56    0
[3]  56    0   64    0   56
[4]   0   56    0   56    0
[5] 304    0   56    0  304
(End)

Links

Crossrefs

Cf. Undirected Hamiltonian paths: A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), A306283 (7 X n), A308131 (n X n).
Cf. A001230, A118067, A306282.

Extensions

a(7) from Guenter Stertenbrink, added by Alex Chernov, Sep 01 2013
a(1)=1, a(2)=0 prepended by Max Alekseyev, Sep 22 2013
a(8) from Alex Chernov, May 10 2014
Name made more precise by Eric W. Weisstein, Apr 14 2019

A118067 Number of (directed) Hamiltonian paths in the 3 X n knight graph.

Original entry on oeis.org

0, 0, 0, 16, 0, 0, 104, 792, 1120, 6096, 21344, 114496, 257728, 1292544, 3677568, 17273760, 46801984, 211731376, 611507360, 2645699504, 7725948608, 32451640000, 97488160384, 397346625760, 1214082434112, 4835168968464, 15039729265856, 58641619298000
Offset: 1

Views

Author

Colin Rose, May 11 2006

Keywords

Comments

1. Jelliss computes the number of tour diagrams (which is equal to half the number of tours). 2. Sequence A079137 computes the number of tour DIAGRAMS for a 4 X k board (again, equal to half the number of tours). 3. Kraitchik (1942) incorrectly reports 376 tour diagrams for the 3 X 8 case; the correct number is 396 (i.e., 792 tours) [cf. Rose, Jelliss].

References

  • Kraitchik, M., Mathematical Recreations. New York: W. W. Norton, pp. 264-5, 1942.

Links

Crossrefs

Cf. A169696, A079137, A083386, A165134.
Cf. A158074. - Eric W. Weisstein, Mar 13 2009

Programs

  • Mathematica
    Mathematica notebook available at: http://www.tri.org.au/knightframe.html

Formula

a(n) = 2 * A169696(n). - Andrew Howroyd, Jul 01 2017

Extensions

a(13) from Eric W. Weisstein, Mar 13 2009
a(14)-a(21) from Seiichi Manyama, Apr 25 2016
a(22)-a(28) from Andrew Howroyd, Jul 01 2017

A169765 Number of closed knight's tour diagrams of a 3 X n chessboard that are symmetric with respect to left-right reflection about a vertical axis.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 24, 0, 0, 0, 276, 0, 0, 0, 2604, 0, 0, 0, 25736, 0, 0, 0, 248816, 0, 0, 0, 2424608, 0, 0, 0, 23581056, 0, 0, 0, 229513584, 0, 0, 0, 2233386048, 0, 0, 0, 21733496960, 0, 0, 0, 211495383968, 0, 0, 0, 2058092298080
Offset: 4

Views

Author

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

Keywords

Examples

			The first example, for n=10, was exhibited by Ernest Bergholt in British Chess Magazine 1918, page 74.

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

A169765(n)=0 unless n mod 4 = 2. And if n mod = 2, A169765(n) = A169765(n) + A169767(n).
Generating function: (2*(2*z^10 - 62*z^18 + 106*z^22 + 624*z^26 - 2560*z^30 - 2464*z^34 + 20640*z^38 + 11112*z^42 - 70304*z^46 - 75840*z^50 + 94976*z^54 + 206528*z^58 - 25216*z^62 - 60672*z^66 - 70656*z^70 - 168960*z^74 + 24576*z^78 + 81920*z^82 + 32768*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).

Extensions

More terms from Alois P. Heinz, Nov 26 2010

A169767 Number of closed knight's tour diagrams of a 3 X n chessboard that have "Eulerian symmetry".

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 124, 0, 0, 0, 1404, 0, 0, 0, 12824, 0, 0, 0, 126696, 0, 0, 0, 1222368, 0, 0, 0, 11930192, 0, 0, 0, 115974192, 0, 0, 0, 1128943296, 0, 0, 0, 10984783168, 0, 0, 0, 106897187552, 0, 0, 0, 1040241749856
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, and the second half of the tour is the same as the first half before rotation. (If the knight starts at one corner, he reaches the opposite corner after 3n/2 moves.)

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

A169767[n]=0 unless n mod 4 = 2.
Generating function: (2*(8*z^14 + 14*z^18 - 182*z^22 - 168*z^26 + 348*z^30 - 1000*z^34 + 13224*z^38 + 22904*z^42 - 105776*z^46 - 111616*z^50 + 292800*z^54 + 217536*z^58 - 294656*z^62 - 114432*z^66 - 22528*z^70 - 44032*z^74 + 180224*z^78 - 65536*z^82 + 32768*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).

A306281 Number of (undirected) Hamiltonian paths on the 6 X n knight graph.

Original entry on oeis.org

0, 0, 0, 744, 18784, 3318960, 389969466, 24964893804, 1770631206422, 143827657320448, 10668015492137018, 763955912402146956, 55382275594728895388, 4008456113318585117624, 285329658478008271167456, 20203324505809248032547768, 1425847547641332606081597198, 100103728161914529291271962728
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), this sequence (6 X n), A306283 (7 X n), A308131 (n X n).

Extensions

a(8)-a(11) from Andrew Howroyd, Oct 14 2019
a(12) from Valentin Gubarev, Dec 23 2024
a(13)-a(18) from Andrew Howroyd, Dec 26 2024
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