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A070030
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Number of closed Knight's tours on a 3 X 2n board.
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19
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0, 0, 0, 0, 16, 176, 1536, 15424, 147728, 1448416, 14060048, 136947616, 1332257856, 12965578752, 126169362176, 1227776129152, 11947846468608, 116266505653888, 1131418872918784, 11010065269439104, 107141489725900544, 1042616896632882688, 10145938076107491328
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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D. E. Knuth, Long and skinny knight's tours, in Selected Papers on Fun and Games, to appear, 2010.
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LINKS
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FORMULA
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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
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
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EXAMPLE
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The smallest 3 X 2n board admitting a closed Knight's tour is the 3 X 10, on which there are 16 such tours.
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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