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A052541
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Expansion of 1/(1-3*x-x^3).
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7
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1, 3, 9, 28, 87, 270, 838, 2601, 8073, 25057, 77772, 241389, 749224, 2325444, 7217721, 22402387, 69532605, 215815536, 669848995, 2079079590, 6453054306, 20029011913, 62166115329, 192951400293, 598883212792, 1858815753705, 5769398661408, 17907079197016
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OFFSET
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0,2
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COMMENTS
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A transform of A000244 under the mapping mapping g(x)->(1/(1-x^3))g(x/(1-x^3)). - Paul Barry, Oct 20 2004
a(n) equals the number of n-length words on {0,1,2,3} such that 0 appears only in a run which length is a multiple of 3. - Milan Janjic, Feb 17 2015
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LINKS
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FORMULA
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G.f.: 1/(1 - 3*x - x^3).
a(n) = 3*a(n-1) + a(n-3), with a(0)=1, a(1)=3.
a(n) = Sum_{alpha = RootOf(-1+3*x+x^3)} (1/15)*(4 + alpha + 2*alpha^2) * alpha^(-n-1).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k) * 3^(n-3*k). - Paul Barry, Oct 20 2004
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MAPLE
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spec := [S, {S=Sequence(Union(Z, Z, Z, Prod(Z, Z, Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..30);
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MATHEMATICA
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CoefficientList[Series[x/(1-3*x-x^3), {x, 0, 30}], x] (* Zerinvary Lajos, Mar 29 2007 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec(x/(1-3*x-x^3)) \\ G. C. Greubel, May 09 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x/(1-3*x-x^3) )); // G. C. Greubel, May 09 2019
(Sage) (x/(1-3*x-x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019
(GAP) a:=[1, 3, 9];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-3]; od; a; # G. C. Greubel, May 09 2019
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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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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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