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URL: https://oeis.org/A099012

⇱ A099012 - OEIS

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A099012
a(n) = 3^(n-1)*Fibonacci(n).
21
0, 1, 3, 18, 81, 405, 1944, 9477, 45927, 223074, 1082565, 5255361, 25509168, 123825753, 601059771, 2917611090, 14162371209, 68745613437, 333698181192, 1619805064509, 7862698824255, 38166342053346, 185263315578333, 899287025215113, 4365230915850336, 21189275974487025
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
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.
E.g.f.: 2 * exp(3*x/2) * sinh(3*sqrt(5)*x/2) / (3*sqrt(5)). - Amiram Eldar, Mar 01 2026
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
(SageMath) [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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 22 2004
STATUS
approved