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A054879
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Closed walks of length 2n along the edges of a cube based at a vertex.
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16
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1, 3, 21, 183, 1641, 14763, 132861, 1195743, 10761681, 96855123, 871696101, 7845264903, 70607384121, 635466457083, 5719198113741, 51472783023663, 463255047212961, 4169295424916643, 37523658824249781, 337712929418248023, 3039416364764232201, 27354747282878089803, 246192725545902808221
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of words of length 2n on alphabet {0,1,2} with an even number (possibly zero) of each letter. - Geoffrey Critzer, Dec 20 2012
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LINKS
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FORMULA
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a(n) = (3^(2*n)+3)/4.
G.f.: 1/4*1/(1-9*x)+3/4*1/(1-x).
E.g.f.: cosh^3(x). O.g.f.: 1/(1-3*1*x/(1-2*2*x/(1-1*3*x))) (continued fraction). - Peter Bala, Nov 13 2006
a(n) = (1/2^3)*Sum_{j = 0..3} binomial(3,j)*(3 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019
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MATHEMATICA
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nn = 40; Select[Range[0, nn]! CoefficientList[Series[Cosh[x]^3, {x, 0, nn}], x], # > 0 &] (* Geoffrey Critzer, Dec 20 2012 *)
Table[(3^(2n)+3)/4, {n, 0, 30}] (* or *) LinearRecurrence[{10, -9}, {1, 3}, 30] (* Harvey P. Dale, Mar 17 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,walk
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AUTHOR
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Paolo Dominici (pl.dm(AT)libero.it), May 23 2000
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STATUS
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approved
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