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A084120
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a(n) = 6*a(n-1) - 3*a(n-2), a(0)=1, a(1)=3.
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12
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1, 3, 15, 81, 441, 2403, 13095, 71361, 388881, 2119203, 11548575, 62933841, 342957321, 1868942403, 10184782455, 55501867521, 302456857761, 1648235544003, 8982042690735, 48947549512401, 266739169002201
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = ((3+sqrt(6))^n + (3-sqrt(6))^n)/2.
G.f.: (1-3*x)/(1-6*x+3*x^2).
E.g.f.: exp(3*x)*cosh(sqrt(6)*x).
a(n) = 3^n * Sum_{k=0..floor(n/2)} C(n, 2*k)*(2/3)^k. - Paul Barry, Sep 10 2005
Lim_{n -> oo} a(n)/a(n-1) = (3 + sqrt(6)) = 5.445489742... - Gary W. Adamson, Mar 19 2008
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(2*k-3)/(x*(2*k-1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
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EXAMPLE
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G.f. = 1 + 3*x + 15*x^2 + 81*x^3 + 441*x^4 + 2403*x^5 + 13095*x^6 + ...
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MATHEMATICA
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LinearRecurrence[{6, -3}, {1, 3}, 30] (* Harvey P. Dale, Feb 25 2014 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polsym(x^2 - 6*x + 3, n)[1+n] / 2)};
(Sage) [lucas_number2(n, 6, 3)/2 for n in range(0, 27)] # Zerinvary Lajos, Jul 08 2008
(Magma) [n le 2 select 3^(n-1) else 6*Self(n-1) -3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 13 2022
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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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