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, 78176337, 126491970, 204668309

Offset: 0

Author

Miklos Kristof, Dec 18 2002

Keywords

• easy
• nonn

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

Examples

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

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).

Crossrefs

Cf. A051597.

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

Programs

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 *)

PARI

a(n) = { 2*fibonacci(n+2) + ((-1)^n - 3)/2 } \\ Harry J. Smith, Mar 14 2010

Python

from sympy import fibonacci
def A066629(n): return (fibonacci(n+2)<<1)-1-(n&1) # Chai Wah Wu, May 05 2025

Formula

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