A022380 Fibonacci sequence beginning 3, 12.

3, 12, 15, 27, 42, 69, 111, 180, 291, 471, 762, 1233, 1995, 3228, 5223, 8451, 13674, 22125, 35799, 57924, 93723, 151647, 245370, 397017, 642387, 1039404, 1681791, 2721195, 4402986, 7124181, 11527167, 18651348, 30178515, 48829863, 79008378

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..4771
• Tanya Khovanova, Recursive Sequences.
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032, A000045, A000285.

Programs

Magma

[-(3/2)*(Fibonacci(n+1)-3*Lucas(n+1)): n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

Mathematica

a[0]=3; a[1] = 12; a[n_]:= a[n-1] +  a[n-2]; Table[a[n],{n,0,30}] (* or *) LinearRecurrence[{1,1},{3,12},31] (* Indranil Ghosh, Feb 19 2017 *)
Table[-(3/2)(Fibonacci[n]-3*LucasL[ n]),{n,40}] (* Harvey P. Dale, Aug 22 2019 *)

Formula

G.f.: (3+9*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = (2^(-n-1)/5)*((15+21*sqrt(5))*(1+sqrt(5))^n + (15-21*sqrt(5))*(1-sqrt(5))^n). - Bogart B. Strauss, Oct 27 2013

a(n) = -(3/2)*(A000045(n+1)-3*A000032(n+1)). - Harvey P. Dale, Aug 22 2019

a(n) = 3*A000285(n). - R. J. Mathar, Jan 08 2020

E.g.f.: 3*(cosh(x/2) + sinh(x/2))*(sqrt(5)*cosh(sqrt(5)*x/2) + 7*sinh(sqrt(5)*x/2))/sqrt(5). - Stefano Spezia, Dec 31 2024