A099012 a(n) = 3^(n-1)*Fibonacci(n).

0, 1, 3, 18, 81, 405, 1944, 9477, 45927, 223074, 1082565, 5255361, 25509168, 123825753, 601059771, 2917611090, 14162371209, 68745613437, 333698181192, 1619805064509, 7862698824255, 38166342053346, 185263315578333

Offset: 0

Author

Paul Barry, Sep 22 2004

Keywords

• easy
• nonn

Comments

Binomial transform is A057088 (with leading 0). Partial sums are A099013. Binomial transform of A015447 (with leading 0).

The ratio a(n+1)/a(n) converges to 3 times the golden ratio (of A000045) as n approaches infinity. - Felix P. Muga II, Mar 10 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014.
• Index entries for linear recurrences with constant coefficients, signature (3,9).

Crossrefs

Cf. A000045, A057088, A099013, A015447.

Programs

Magma

[3^(n-1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

Mathematica

a[n_]:=(MatrixPower[{{1,5},{1,-4}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Table[3^(n-1) Fibonacci[n],{n,0,30}] (* or *) LinearRecurrence[{3,9},{0,1},30] (* Harvey P. Dale, Nov 07 2017 *)

PARI

a(n)=3^(n-1)*fibonacci(n) \\ Charles R Greathouse IV, Sep 24 2015

Sage

[lucas_number1(n,3,-9) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009

Formula

G.f.: x/(1 - 3*x - 9*x^2).

a(n) = 3*a(n-1) + 9*a(n-2).

a(n) = sqrt(5)(3/2 + 3*sqrt(5)/2)^n/15 - sqrt(5)*(3/2 - 3*sqrt(5)/2)^n/15.