A122069 a(n) = 3*a(n-1) + 9*a(n-2) for n > 1, with a(0)=1, a(1)=3.

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

Philippe Deléham, Oct 15 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,9).

Crossrefs

Third row of A234357.

Cf. A000045, A085504, A099012.

Programs

GAP

List([0..25], n-> 3^n*Fibonacci(n+1) ); # G. C. Greubel, Oct 03 2019

Magma

[3^n*Fibonacci(n+1): n in [0..25]]; // G. C. Greubel, Oct 03 2019

Maple

with(combinat); seq(3^n*fibonacci(n+1), n=0..25); # G. C. Greubel, Oct 03 2019

Mathematica

Table[3^n*Fibonacci[n+1], {n,0,25}] (* G. C. Greubel, Oct 03 2019 *)
LinearRecurrence[{3,9},{1,3},30] (* Harvey P. Dale, Apr 28 2020 *)

PARI

vector(26, n, 3^(n-1)*fibonacci(n) ) \\ G. C. Greubel, Oct 03 2019

Sage

[lucas_number1(n,3,-9) for n in range(1, 23)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = 3^n*Fibonacci(n+1) = 3^n*A000045(n+1).

a(n) = Sum_{k=0..n} 2^k*A016095(n,k).

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

Limit_{n->oo} a(n+1)/a(n) = 3*(1+sqrt(5))/2.

a(n) = A099012(n+1). - R. J. Mathar, Aug 02 2008

a(n) = A085504(n) for n >= 2. - Georg Fischer, Nov 03 2018

Extensions

Corrected by T. D. Noe, Nov 07 2006