A084120 a(n) = 6*a(n-1) - 3*a(n-2), a(0)=1, a(1)=3.

1, 3, 15, 81, 441, 2403, 13095, 71361, 388881, 2119203, 11548575, 62933841, 342957321, 1868942403, 10184782455, 55501867521, 302456857761, 1648235544003, 8982042690735, 48947549512401, 266739169002201

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.: (1-3*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

a(n) = Sum_{k=0..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

a(n) = A138395(n) - 3*A138395(n-1). - R. J. Mathar, May 11 2022

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
(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

Crossrefs

Cf. A084059, A138395, A147720.

Sequence in context: A198628 A233020 A246020 * A163470 A122868 A264225

Adjacent sequences: A084117 A084118 A084119 * A084121 A084122 A084123

Keyword

easy,nonn

Author

Paul Barry, May 13 2003

Status

approved