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
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).
Programs
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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).
Comments