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A052925
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Expansion of (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).
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2
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2, 2, 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369, 121394, 317812, 832041, 2178310, 5702888, 14930353, 39088170, 102334156, 267914297, 701408734, 1836311904, 4807526977, 12586269026, 32951280100, 86267571273
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OFFSET
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0,1
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LINKS
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FORMULA
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G.f.: (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), with a(0)=2, a(1)=2, a(2)=4, a(3)=9.
a(n) = 1 + Sum_{alpha=RootOf(1-3*z+z^2)} (1/5)*(2-3*alpha)*alpha^(-1-n).
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MAPLE
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spec:=[S, {S=Union(Sequence(Z), Sequence(Prod(Sequence(Z), Sequence(Z), Z) ))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(coeff(series((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 17 2019
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MATHEMATICA
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CoefficientList[Series[(-2+6*x-4*x^2+x^3)/(-1+x)/(1-3*x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 22 2012 *)
LinearRecurrence[{4, -4, 1}, {2, 2, 4, 9}, 30] (* G. C. Greubel, Oct 17 2019 *)
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PROG
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(Magma) I:=[2, 2, 4, 9]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2) +Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012
(PARI) my(x='x+O('x^30)); Vec((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2))) \\ G. C. Greubel, Oct 17 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2))).list()
(GAP) a:=[2, 4, 9];; for n in [4..30] do a[n]:=4*a[n-1]-4*a[n-2]+a[n-3]; od; Concatenation([2], a); # G. C. Greubel, Oct 17 2019
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CROSSREFS
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Apart from first term, same as A055588.
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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