A141041 a(n) = ((3 + 2*sqrt(3))^n + (3 - 2*sqrt(3))^n)/2.

1, 3, 21, 135, 873, 5643, 36477, 235791, 1524177, 9852435, 63687141, 411680151, 2661142329, 17201894427, 111194793549, 718774444575, 4646231048097, 30033709622307, 194140950878133, 1254946834135719, 8112103857448713

Offset

0,2

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*abs(A099842(n-1)), for n > 0.

G.f.: (1-3*x)/(1-6*x-3*x^2). - Philippe Deléham, Mar 03 2012

a(n) = 6*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 3. - Philippe Deléham, Mar 03 2012

a(n) = Sum_{k=0..n} A201701(n,k)*3^(n-k). - Philippe Deléham, Mar 03 2012

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(4*k-3)/(x*(4*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013

a(n) = (-i*sqrt(3))^n * ChebyshevT(n, i*sqrt(3)). - G. C. Greubel, Oct 10 2022

Mathematica

a[n_]= ((3+2*Sqrt[3])^n + (3-2*Sqrt[3])^n)/2; Table[FullSimplify[a[n]], {n, 0, 30}]
LinearRecurrence[{6, 3}, {1, 3}, 30] (* Harvey P. Dale, Aug 25 2014 *)

Prog

(Magma) [n le 2 select 3^(n-1) else 6*Self(n-1) +3*Self(n-2): n in [1..31]]; // G. C. Greubel, Oct 10 2022
(SageMath)
A141041 = BinaryRecurrenceSequence(6, 3, 1, 3)
[A141041(n) for n in range(31)] # G. C. Greubel, Oct 10 2022

Crossrefs

Cf. A011943, A034478, A081336, A099842, A201701.

Sequence in context: A333030 A125701 A124812 * A079753 A346935 A137969

Adjacent sequences: A141038 A141039 A141040 * A141042 A141043 A141044

Keyword

nonn

Author

Roger L. Bagula, Aug 18 2008

Extensions

Edited by N. J. A. Sloane, Aug 24 2008

Status

approved