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

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

1,10

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

Seiichi Manyama, Table of n, a(n) for n = 1..2031 (terms 1..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).

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).

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 *)

Crossrefs

Cf. A070030, A169696, A169765-A169777.

Sequence in context: A135925 A188784 A123935 * A002607 A221404 A221726

Adjacent sequences: A169761 A169762 A169763 * A169765 A169766 A169767

Keyword

nonn,easy

Author

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

Status

approved