A062114 a(n) = 2*Fibonacci(n) - (1 - (-1)^n)/2.

0, 1, 2, 3, 6, 9, 16, 25, 42, 67, 110, 177, 288, 465, 754, 1219, 1974, 3193, 5168, 8361, 13530, 21891, 35422, 57313, 92736, 150049, 242786, 392835, 635622, 1028457, 1664080, 2692537, 4356618, 7049155, 11405774, 18454929, 29860704, 48315633, 78176338, 126491971

Offset: 0

Author

Olivier Gérard, Jun 05 2001

Keywords

• easy
• nonn

Examples

			a(4) = a(3) + a(2) + (1+1)/2 = 3 + 2 + 1 = 6.
G.f. = x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 25*x^7 + ... - _Michael Somos_, Oct 17 2018
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4785 (first 401 terms from Harry J. Smith)
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Crossrefs

Cf. A000045, A052952, A134513.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x-x^2)/((1-x)*(1+x)*(1-x-x^2)))); // G. C. Greubel, Oct 16 2018

Maple

A062114 := proc(n)
    2*combinat[fibonacci](n)-(1-(-1)^n)/2 ;
end proc: # R. J. Mathar, Aug 12 2012
# second Maple program:
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-1|2|1>>^n.<<[$0..3][]>>)[1$2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 01 2018

Mathematica

Join[{a=0,b=1},Table[If[EvenQ[a],c=a+b+1,c=a+b];a=b;b=c,{n,0,5!}]](* Vladimir Joseph Stephan Orlovsky, Jan 10 2011 *)
Table[2Fibonacci[n]-(1-(-1)^n)/2,{n,0,40}] (* or *) LinearRecurrence[ {1,2,-1,-1},{0,1,2,3},41] (* Harvey P. Dale, Nov 02 2011 *)

PARI

{ h=-1; g=1; for (n=0, 400, f=g + h; h=g; g=f; write("b062114.txt", n, " ", 2*f - (1 - (-1)^n)/2) ) } \\ Harry J. Smith, Aug 01 2009

PARI

x='x+O('x^30); concat([0], Vec(x*(1+x-x^2)/((1-x)*(1+x)*(1-x-x^2) ))) \\ G. C. Greubel, Oct 16 2018

Formula

A bistable recurrence; Fibonacci with a grain of salt: a(0)=0; a(1)=1; a(n) = a(n-1) + a(n-2) + (1 + (-1)^n)/2.

a(n+1) = Sum_{k=0..n} binomial(n-floor(k/2), floor(k/2)). - Benoit Cloitre, May 05 2005

Starting with 1, equals row sums of triangle A134513. - Gary W. Adamson, Oct 28 2007

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4), n > 3. - Harvey P. Dale, Nov 02 2011

G.f.: x*(1+x-x^2)/( (1-x)*(1+x)*(1-x-x^2) ). - R. J. Mathar, Aug 12 2012

a(n) = -(-1)^n * a(-n) for all n in Z. - Michael Somos, Oct 17 2018

Extensions

Definition corrected by Harry J. Smith, Aug 01 2009