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A052953
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Expansion of 2*(1-x-x^2)/((1-x)*(1+x)*(1-2*x)).
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7
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2, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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G.f.: 2*(1-x-x^2)/((1-x^2)*(1-2*x)).
a(n) = a(n-1) + 2*a(n-2) - 2.
a(n) = 1 + Sum_{alpha=RootOf(-1+z+2*z^2)} (1 + 4*alpha)*alpha^(-1-n)/9.
a(2n) = 2*a(n-1)-2, a(2n+1) = 2*a(2n). - Lee Hae-hwang, Oct 11 2002
a(n) = (2^(n+1) - (-1)^(n+1))/3 + 1. (End)
E.g.f.: (2*exp(2*x) + 3*exp(x) + exp(-x))/3. - G. C. Greubel, Oct 21 2019
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MAPLE
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spec:= [S, {S=Union(Sequence(Union(Prod(Union(Z, Z), Z), Z)), Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq((2^(n+1) +3 +(-1)^n)/3, n=0..40); # G. C. Greubel, Oct 21 2019
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MATHEMATICA
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LinearRecurrence[{2, 1, -2}, {2, 2, 4}, 40] (* G. C. Greubel, Oct 22 2019 *)
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PROG
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(Sage) [(2^(n+1) +3 +(-1)^n)/3 for n in (0..40)] # G. C. Greubel, Oct 21 2019
(PARI) vector(41, n, (2^n +3 -(-1)^n)/3 ) \\ G. C. Greubel, Oct 21 2019
(Magma) [(2^(n+1) +3 +(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Oct 21 2019
(GAP) List([0..40], n-> (2^(n+1) +3 +(-1)^n)/3); # G. C. Greubel, Oct 21 2019
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CROSSREFS
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Apart from initial term, equals A026644(n+1) + 2.
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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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