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A084120 a(n)=6a(n-1)-3a(n-2), a(0)=1,a(1)=3. 11
1, 3, 15, 81, 441, 2403, 13095, 71361, 388881, 2119203, 11548575, 62933841, 342957321, 1868942403, 10184782455, 55501867521, 302456857761, 1648235544003, 8982042690735, 48947549512401, 266739169002201 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A084059.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-3).

FORMULA

a(n)=((3+sqrt(6))^n+(3-sqrt(6))^n)/2; G.f.: (1-3x)/(1-6x+3x^2); E.g.f.: exp(3x)cosh(sqrt(6)x).

a(n)=3^n*sum{k=0..floor(n/2), C(n, 2k)(2/3)^k}; - Paul Barry, Sep 10 2005

a(n)/a(n-1) tends to (3 + sqrt(6)) = 5.445489742... - Gary W. Adamson, Mar 19 2008

a(n)=Sum_{k, 0<=k<=n}A147720(n,k)*3^k. - Philippe Deléham, Nov 15 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

EXAMPLE

G.f. = 1 + 3*x + 15*x^2 + 81*x^3 + 441*x^4 + 2403*x^5 + 13095*x^6 + ...

MATHEMATICA

LinearRecurrence[{6, -3}, {1, 3}, 30] (* Harvey P. Dale, Feb 25 2014 *)

PROG

(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

CROSSREFS

Cf. A138395.

Sequence in context: A198628 A233020 A246020 * A163470 A122868 A264225

Adjacent sequences:  A084117 A084118 A084119 * A084121 A084122 A084123

KEYWORD

easy,nonn,changed

AUTHOR

Paul Barry, May 13 2003

STATUS

approved

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Last modified December 10 23:29 EST 2019. Contains 329910 sequences. (Running on oeis4.)