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A121961
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Expansion of g.f.: 1/( (1+2*x)*(1-2*x-4*x^2)*(1-2*x^2)^2 ).
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1
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1, 0, 12, 8, 108, 160, 960, 2144, 9040, 24832, 89664, 270976, 916416, 2885120, 9500160, 30412288, 99084544, 319299584, 1035979776, 3347073024, 10842246144, 35064422400, 113514577920, 367253348352, 1188632055808, 3846143410176, 12447083347968, 40278203727872
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OFFSET
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0,3
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COMMENTS
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Based on characteristic polynomial of a square-within-a-square bonding graph.
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LINKS
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FORMULA
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a(2*n) = 4^(n+1)*(Fibonacci(2*n-2) + 1) - 2^n*(3*n-1).
a(2*n+1) = 2^(2*n+3)*(Fibonacci(2*n-1) - 1) + 2^(n+2)*n. (End)
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MAPLE
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seq(coeff(series(1/((1-8*x^2-8*x^3)*(1-2*x^2)^2), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 05 2019
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MATHEMATICA
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M = {{0, 1, 0, 1, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 1, 1, 0}, {1, 0, 1, 0, 0, 0, 1, 1}, {1, 1, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0, 0, 0}}; f[x_] = Det[M - x*IdentityMatrix[8]]; Table[ SeriesCoefficient[ Series[x/(x^10*f[1/x]), {x, 0, 30}], n], {n, 0, 30}]
LinearRecurrence[{0, 12, 8, -36, -32, 32, 32}, {1, 0, 12, 8, 108, 160, 960}, 40] (* Harvey P. Dale, May 28 2017 *)
Table[If[EvenQ[n], 4^(n/2 +1)*(Fibonacci[n-2] +1) - 2^(n/2)*(3*n/2 -1), 2^(n+2)*(Fibonacci[n-2] -1) + 2^((n+1)/2)*(n-1)], {n, 0, 40}] (* G. C. Greubel, Oct 05 2019 *)
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PROG
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(PARI) my(x='x+O('x^40)); Vec(1/((1-8*x^2-8*x^3)*(1-2*x^2)^2)) \\ G. C. Greubel, Oct 05 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-8*x^2-8*x^3)*(1-2*x^2)^2) )); // G. C. Greubel, Oct 05 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-8*x^2-8*x^3)*(1-2*x^2)^2)).list()
(GAP) a:=[1, 0, 12, 8, 108, 160, 960];; for n in [8..40] do a[n]:=12*a[n-2] +8*a[n-3]-36*a[n-4]-32*a[n-5]+32*a[n-6]+32*a[n-7]; od; a; # G. C. Greubel, Oct 05 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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