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A099783
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a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k)*3^(n-2*k).
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6
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1, 3, 9, 30, 108, 405, 1548, 5967, 23085, 89451, 346842, 1345248, 5218263, 20242872, 78528609, 304640595, 1181814705, 4584708702, 17785841652, 68998115709, 267670245492, 1038395956527, 4028337876861, 15627474388899, 60624993311226
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OFFSET
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0,2
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COMMENTS
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In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).
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LINKS
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FORMULA
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G.f.: (1-3*x)/((1-3*x)^2 - 3*x^3).
a(n) = 6*a(n-1) - 9*a(n-2) + 3*a(n-3).
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MAPLE
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seq(coeff(series((1-3*x)/((1-3*x)^2 - 3*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019
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MATHEMATICA
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LinearRecurrence[{6, -9, 3}, {1, 3, 9}, 30] (* G. C. Greubel, Sep 04 2019 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((1-3*x)/((1-3*x)^2 - 3*x^3)) \\ G. C. Greubel, Sep 04 2019
(Magma) I:=[1, 3, 9]; [n le 3 select I[n] else 6*Self(n-1) - 9*Self(n-2) + 3*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 04 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-3*x)/((1-3*x)^2 - 3*x^3)).list()
(GAP) a:=[1, 3, 9];; for n in [4..30] do a[n]:=6*a[n-1]-9*a[n-2]+3*a[n-3]; od; a; # G. C. Greubel, Sep 04 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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STATUS
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approved
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