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A144045
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The number of paths of a chess Rook in a cube, from (1,1,1) to (n,n,n), where the rook may move in steps that are multiples of (1,0,0), (0,0,1), or (0,0,1).
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1
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1, 6, 222, 9918, 486924, 25267236, 1359631776, 75059524392, 4223303759148, 241144782230124, 13930829740017132, 812470444305924300, 47760356825349969600, 2826309951801018736800, 168207011284961649886800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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LINKS
| M. Kauers and D. Zeilberger, The Computational Challenge of Enumerating High-Dimensional Rook Walks
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FORMULA
| a(n) satisfies the recurrence relation a(1) = 1; a(2) = 6; a(3) = 222; a(4) = 9918; a(n) = ((-121 n^3 + 575n^2 - 872n + 412)a(n - 1) + (-475n^3 + 4887n^2 - 16202n + 17448)a(n - 2) + (1746n^3 - 19818n^2 + 75060n - 94896)a(n - 3) + (-1152n^3 + 16128n^2 - 74880n + 115200)a(n - 4))/(-(2n^3 - 8n^2 + 10n - 4)), n>= 5.
G.f. 1+int(6*hypergeom([1/3, 2/3],[2],27*x*(3*x-2)/(4*x-1)^3)/((4*x-1)*(64*x-1)),x) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Dec 10 2009]
Asymptotic (Frederic Chyzak, 2010): 9*sqrt(3)/(40*pi*n)*64^(n-1)
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EXAMPLE
| a(2)=6 because there are 6 Rook paths from (1,1,1) to (2,2,2).
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CROSSREFS
| Cf. A051708
Sequence in context: A015477 A144658 A203637 * A061610 A054324 A117255
Adjacent sequences: A144042 A144043 A144044 * A144046 A144047 A144048
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KEYWORD
| nonn
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AUTHOR
| Martin J. Erickson (erickson(AT)truman.edu), Sep 08 2008
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