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A092812
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Number of closed walks of length 2*n on the 4-cube.
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8
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1, 4, 40, 544, 8320, 131584, 2099200, 33562624, 536903680, 8590065664, 137439477760, 2199025352704, 35184380477440, 562949986975744, 9007199388958720, 144115188612726784, 2305843011361177600
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OFFSET
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0,2
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COMMENTS
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With interpolated zeros this has a(n) = (6*0^n + 4^n + (-4)^n + 4*2^n + 4*(-2)^n)/16 and counts closed walks of length n at a vertex of the 4-cube. [Typo corrected by Alexander R. Povolotsky, May 26 2008]
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LINKS
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FORMULA
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G.f.: (1-16*x+24*x^2)/((1-4*x)*(1-16*x)).
a(n) = 3*0^n/8 + 16^n/8 + 4^n/2.
E.g.f.: cosh^4(x).
O.g.f.: 1/(1-4*1*x/(1-3*2*x/(1-2*3*x/(1-1*4*x)))) (continued fraction). (End)
a(n) = 20*a(n-1) - 64*a(n-2); a(0) = 1, a(1) = 4, a(2) = 40. - Harvey P. Dale, Aug 23 2011
a(n) = (1/2^4)*Sum_{j = 0..4} binomial(4, j)*(4 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019
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MATHEMATICA
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CoefficientList[Series[(1-16x+24x^2)/((1-4x)(1-16x)), {x, 0, 30}], x] (* or *) Join[{1}, LinearRecurrence[{20, -64}, {4, 40}, 30]] (* Harvey P. Dale, Aug 23 2011 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,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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