login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A052952 a(n) = Fibonacci(n+2) - (1-(-1)^n)/2. 32
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, 63245985, 102334155 (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

LINKS

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1023

K. Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1, eq (1)

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.

H. Ohtsuka and S. Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 153-159.

Index entries for 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_{j=0..k} F(2*j+1) = F(2*(k+1)) for k >= 0; a(2*k-1) = Sum_{j=0..k} F(2*j) = F(2*k+1)-1 for k >= 1 (F = A000045, 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) with n>1, a(0)=a(1)=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+1) = a(n) + a(n-1) + A059841(n+1). - Reinhard Zumkeller, Jan 06 2012

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

a(n) = floor(phi*Fibonacci(n+1)), where phi is the golden section. - Michel Dekking, Dec 02 2016

a(n) = -(-1)^n * a(-4-n) for all n in Z. - Michael Somos, Dec 03 2016

a(n) = Sum_{k=0..n} Sum_{i=0..n} C(n-k-1,k-i). - Wesley Ivan Hurt, Sep 21 2017

a(n) = floor(1/(Sum_{k>=n+4} 1/Fibonacci(k))) [Ohtsuka and Nakamura]. - Michel Marcus, Aug 09 2018

EXAMPLE

G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 33*x^7 + ...

MAPLE

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..40) ; # N. J. A. Sloane, May 25 2008

MATHEMATICA

Table[Fibonacci[n+2] -(1-(-1)^n)/2, {n, 0, 40}] (* Vincenzo Librandi, Dec 02 2016 *)

PROG

(PARI) {a(n) = 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

(MAGMA) [Fibonacci(n+2)-(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Dec 02 2016

(Sage) [fibonacci(n+2) -(1-(-1)^n)/2 for n in (0..40)] # G. C. Greubel, Jul 10 2019

(GAP) List([0..40], n-> Fibonacci(n+2) -(1-(-1)^n)/2); # G. C. Greubel, Jul 10 2019

CROSSREFS

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

Cf. A062114, A173284, A059841, A014217.

Partial sums of A008346.

Cf. A000045.

Sequence in context: A147622 A173534 A074331 * A245121 A153339 A275989

Adjacent sequences:  A052949 A052950 A052951 * A052953 A052954 A052955

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

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

Better description from Olivier Gérard, Jun 05 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 13 18:57 EDT 2019. Contains 327981 sequences. (Running on oeis4.)