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

0,3

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).

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.

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 *)

Prog

(Sage) [lucas_number1(n, 3, -9) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009
(Magma) [3^(n-1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011
(PARI) a(n)=3^(n-1)*fibonacci(n) \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A000045, A057088, A099013, A015447.

Sequence in context: A337193 A036290 A078904 * A122069 A103897 A119424

Adjacent sequences: A099009 A099010 A099011 * A099013 A099014 A099015

Keyword

easy,nonn

Author

Paul Barry, Sep 22 2004

Status

approved