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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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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

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
Showing 1-2 of 2 results.

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