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A052952 Fibonacci(n+2) - (1-(-1)^n)/2. 30
1, 1, 3, 4, 8, 12, 21, 33, 55, 88, 144, 232, 377, 609, 987, 1596, 2584, 4180, 6765, 10945, 17711, 28656, 46368, 75024, 121393, 196417, 317811, 514228, 832040, 1346268, 2178309, 3524577, 5702887, 9227464, 14930352, 24157816, 39088169 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equals row sums of triangle A173284. - Gary W. Adamson, Feb 14 2010

The Kn21 sums, see A180662 for the definition of these sums, of the ‘Races with Ties’ triangle A035317 equal this sequence. - Johannes W. Meijer, Jul 20 2011

a(n+1) = a(n) + a(n-1) + A059841(n+1). - Reinhard Zumkeller, Jan 06 2012

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1023

Steven Linton, James Propp, Tom Roby, Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.

Index to sequences with linear recurrences with constant coefficients, signature (1,2,-1,-1).

FORMULA

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

a(n) = A074331(n+1).

Recurrence: a(0)=1, a(1)=1, a(2)=3, a(n) = 2*a(n-2) + a(n-3) + 1.

Sum(1/5*(3+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+_Z^2)) + Sum(-1/2*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2)).

a(2*k)=sum(F(2*j+1), j=0..k)= F(2*(k+1)), k >= 0; a(2*k-1)=sum(F(2*j), j=0..k)= F(2*k+1)-1, k >= 1; F(n)=A000045(n) (Fibonacci numbers).

a(n) = a(n-1) + a(n-2) + (1+(-1)^n)/2.

a(n) = sum{k=0..floor(n/2), binomial(n-k+1, k)}. - Paul Barry, Oct 23 2004

a(n) = floor(phi^(n+2) / sqrt(5)), where phi is the golden ratio: phi = (1+sqrt(5))/2. - Reinhard Zumkeller, Apr 19 2005

a(n) = fibonacci(n-1)+a(n-2), a(2)=1. - Zerinvary Lajos, Mar 17 2008

a(n) = floor(fibonacci(n+3)^2/fibonacci(n+4)). - Gary Detlefs Nov 29 2010

a(n) = (A001595(n+3)-A066983(n+4))/2. - Gary Detlefs Dec 19 2010

a(4*n) = F(4*n+2); a(4*n+1) = F(4*n+3) - 1; a(4*n+2) = F(4*n+4); a(4*n+3) = F(4*n+5) - 1. - Johannes W. Meijer, Jul 20 2011

a(n) = floor(|F((1+I)*(n+2))|), n>=0, with the complex Fibonacci function F: C -> C, z -> F(z) with

  F(z) := (exp(log(phi)*z) - exp(I*Pi*z)*exp(-log(phi)*z))/(2*phi-1) with the modulus |z|, the imaginary unit I and the golden section phi:=(1+sqrt(5))/2. A Conjecture: For F(z) see, e.g., the T. Koshy reference. ch. 45, p. 523, where F is called f, given in A000045. - Wolfdieter Lang, Jul 24 2012

5*a(n) = (L(n+3)-1)*(L(n+4)+3) -14 -sum_{k=0..n} L(k+1)*L(k+5) = (L(n+3)-1)*(L(n+4)+3) -L(2*n+7) +A168309(n), where L=A000032. - J. M. Bergot, Jun 13 2014

MAPLE

spec := [S, {S=Prod(Sequence(Union(Prod(Z, Z), Z)), Sequence(Prod(Z, Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20); # Zerinvary Lajos, Mar 17 2008

A052952 :=proc(n)

    option remember;

    local t1;

    if n <= 1 then

        return 1 ;

    fi:

    if n mod 2 = 1 then

        t1:=0

    else

        t1:=1;

    fi:

    procname(n-1)+procname(n-2)+t1;

end proc;

seq(A052952(n), n=0..100) ; # N. J. A. Sloane, May 25 2008

MATHEMATICA

a=0; b=0; lst={a, b}; Do[z=a+b+1; AppendTo[lst, z]; a=b; b=z; z=a+b; AppendTo[lst, z]; a=b; b=z, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)

PROG

(PARI) a(n)=if(n<0, 0, fibonacci(n+2)-n%2)

(Haskell)

a052952 n = a052952_list !! n

a052952_list = 1 : 1 : zipWith (+)

   a059841_list (zipWith (+) a052952_list $ tail a052952_list)

-- Reinhard Zumkeller, Jan 06 2012

CROSSREFS

a(n) = A054450(n+1, 1) (second column of triangle).

Cf. A062114, A173284, A059841, A014217.

Sequence in context: A147622 A173534 * A074331 A245121 A153339 A033955

Adjacent sequences:  A052949 A052950 A052951 * A052953 A052954 A052955

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

Additional formulae and more terms from Wolfdieter Lang, May 02 2000

Better description from Olivier Gérard, Jun 05 2001

More terms from James A. Sellers, Jun 06 2000

STATUS

approved

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Last modified October 23 00:52 EDT 2014. Contains 248411 sequences.