A097134 a(n) = 3*Fibonacci(2*n) + 0^n.

1, 3, 9, 24, 63, 165, 432, 1131, 2961, 7752, 20295, 53133, 139104, 364179, 953433, 2496120, 6534927, 17108661, 44791056, 117264507, 307002465, 803742888, 2104226199, 5508935709, 14422580928, 37758807075, 98853840297, 258802713816

Offset

0,2

Comments

Binomial transform of A097133.

Image of 1/(1-3x) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005

Links

Table of n, a(n) for n=0..27.

Index entries for linear recurrences with constant coefficients, signature (3,-1).

Formula

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

a(n) = 3*a(n-1) - a(n-2) for n > 2.

a(n) = Sum_{k=0..n} binomial(n, k)*(3*Fibonacci(k)+(-1)^k).

a(n) = A097135(2*n).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k)*(-1)^k*3^(n-2*k). - Paul Barry, Jan 16 2005

a(n) = Fibonacci(n+2)^2 - Fibonacci(n-2)^2. - Gary Detlefs, Dec 03 2010

a(n) = Fibonacci(6*n) - 5*Fibonacci(2*n)^3 for n > 0. - Gary Detlefs, Oct 18 2011

E.g.f.: 1 + 6*exp(3*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Aug 19 2019

Prog

(Magma) [3*Fibonacci(2*n)+0^n: n in [0..30]]; // Vincenzo Librandi, Apr 21 2011
(PARI) a(n)=3*fibonacci(n+n)+0^n \\ Charles R Greathouse IV, Oct 18 2011

Crossrefs

Cf. A000045.

Sequence in context: A090400 A123888 A166290 * A123892 A269531 A064831

Adjacent sequences: A097131 A097132 A097133 * A097135 A097136 A097137

Keyword

nonn,easy

Author

Paul Barry, Jul 26 2004

Status

approved