A066629 a(n) = 2*Fibonacci(n+2) + ((-1)^n - 3)/2.

1, 2, 5, 8, 15, 24, 41, 66, 109, 176, 287, 464, 753, 1218, 1973, 3192, 5167, 8360, 13529, 21890, 35421, 57312, 92735, 150048, 242785, 392834, 635621, 1028456, 1664079, 2692536, 4356617, 7049154, 11405773, 18454928, 29860703, 48315632

Offset

0,2

Comments

Fibonacci-like numbers made from Asher Auel's triangle A(n,m) (A051597) satisfying A(0,0)=1, A(1,0)=2, A(1,1)=2, etc..: then a(0)=1, a(1)=2, a(n) = A(n,0) + A(n-1,1) + A(n-2,2) + ...

Equals row sums of triangle A153864. - Gary W. Adamson, Jan 03 2009

Links

Harry J. Smith, Table of n, a(n) for n = 0..250

Index entries for linear recurrences with constant coefficients, signature (1, 2, -1, -1).

Formula

Lim_{n->inf} a(n)/a(n-1) = (1+sqrt(5))/2. If n even: a(n) = a(n-1) + a(n-2) + 2; if n odd: a(n) = a(n-1) + a(n-2) + 1.

G.f.: (1+x+x^2)/((1-x-x^2)(1-x)(1+x)). - R. J. Mathar, Sep 19 2008

a(0)=1, a(1)=2, a(2)=5, a(3)=8, a(n)=a(n-1)+2*a(n-2)-a(n-3)-a(n-4). - Harvey P. Dale, Oct 09 2011

Example

a(5) = A(5,0) + A(4,1) + A(3,2) = 6 + 11 + 7 = 24.

Maple

A066629 := proc(n)
    2*combinat[fibonacci](n+2)+((-1)^n-3)/2 ;
end proc:
seq(A066629(n), n=0..10) ; # R. J. Mathar, Apr 13 2016

Mathematica

Join[{b=1}, a=0; Table[If[OddQ[a]&&EvenQ[b], c=a+b+2, c=a+b+1]; a=b; b=c, {n, 0, 5!}]] (* Vladimir Joseph Stephan Orlovsky, Jan 10 2011 *)
Table[2Fibonacci[n+2]+((-1)^n-3)/2, {n, 0, 40}] (* or *) LinearRecurrence[ {1, 2, -1, -1}, {1, 2, 5, 8}, 41] (* Harvey P. Dale, Oct 09 2011 *)

Prog

(PARI) print1(y=1, ", ", z=2, ", "); for(n=2, 35, print1(a=z+y+2-n%2, ", "); y=z; z=a)
(PARI) { for (n=0, 250, a=2*fibonacci(n+2) + ((-1)^n - 3)/2; write("b066629.txt", n, " ", a) ) } \\ Harry J. Smith, Mar 14 2010

Crossrefs

Cf. A051597.

Cf. A153864. - Gary W. Adamson, Jan 03 2009

Sequence in context: A309662 A066897 A078697 * A154327 A074027 A018156

Adjacent sequences: A066626 A066627 A066628 * A066630 A066631 A066632

Keyword

easy,nonn

Author

Miklos Kristof, Dec 18 2002

Status

approved