A085447 a(n) = 6*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=6.

2, 6, 38, 234, 1442, 8886, 54758, 337434, 2079362, 12813606, 78960998, 486579594, 2998438562, 18477210966, 113861704358, 701647437114, 4323746327042, 26644125399366, 164188498723238, 1011775117738794

Offset: 0

Author

Gary W. Adamson, Jul 01 2003

Keywords

• nonn
• easy

Comments

a(n+1)/a(n) converges to 3 + sqrt 10.

a(0)/a(1) = 1/3 = [3]; a(1)/a(2) = 6/38 = [6,3]; a(2)/a(3) = 38/234 = [6,6,3], a(3)/a(4) = 234/1442 = [6,6,6,3]; a(4)/a(5) = 1442/8886 = [6,6,6,6,3];...etc. Lim a(n)/a(n+1) as n approaches infinity = 0.1622776...= 1/(3 + sqrt10) = sqrt(10) - 3.

Examples

			a(4) = 6*a(3)+a(2) = 6*234+38 = 1442.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{6,n}
• Tanya Khovanova, Recursive Sequences
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for linear recurrences with constant coefficients, signature (6,1).

Crossrefs

Cf. A005667, A005668.

Programs

Magma

I:=[2,6]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

Mathematica

RecurrenceTable[{a[0] == 2, a[1] == 6, a[n] == 6 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{6,1}, {2,6}, 30] (* G. C. Greubel, Nov 07 2018 *)

PARI

x='x+O('x^30); Vec(2*(1-3*x)/(1-6*x-x^2)) \\ G. C. Greubel, Nov 07 2018

Formula

a(n) = (3 + sqrt 10)^n + (3 - sqrt 10)^n = A005668(n+1) + A005668(n-1).

O.g.f.: 2*(-1+3*x)/(-1+6*x+x^2). - R. J. Mathar, Dec 02 2007

a(n) = 2*A005667(n). - R. J. Mathar, Nov 10 2013

Extensions

Edited and extended by Henry Bottomley, Jul 13 2003