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A070030 Number of closed Knight's tours on a 3 X 2n board. 19
0, 0, 0, 0, 16, 176, 1536, 15424, 147728, 1448416, 14060048, 136947616, 1332257856, 12965578752, 126169362176, 1227776129152, 11947846468608, 116266505653888, 1131418872918784, 11010065269439104, 107141489725900544 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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).

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

N. D. Elkies and R. P. Stanley, The mathematical knight, Math. Intelligencer, 25 (No. 1) (2003), 22-34.

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).

Eric Weisstein's World of Mathematics, Knight's Tour

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

EXAMPLE

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

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 *)

PROG

(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))

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.

Sequence in context: A021129 A268869 A268459 * A283278 A231127 A007144

Adjacent sequences:  A070027 A070028 A070029 * A070031 A070032 A070033

KEYWORD

easy,nice,nonn

AUTHOR

Noam D. Elkies, Apr 13 2002

EXTENSIONS

Comment corrected by Johannes W. Meijer, Nov 22 2010

STATUS

approved

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Last modified December 10 20:48 EST 2017. Contains 295856 sequences.