A167808 Numerator of x(n), where x(n) = x(n-1) + x(n-2) with x(0)=0, x(1)=1/2.

0, 1, 1, 1, 3, 5, 4, 13, 21, 17, 55, 89, 72, 233, 377, 305, 987, 1597, 1292, 4181, 6765, 5473, 17711, 28657, 23184, 75025, 121393, 98209, 317811, 514229, 416020, 1346269, 2178309, 1762289, 5702887, 9227465, 7465176, 24157817, 39088169, 31622993

Offset: 0

Author

Reinhard Zumkeller, Nov 12 2009

Keywords

• nonn
• frac

Comments

Define a sequence c(n) by c(0)=0, c(1)=1; thereafter c(n) = (c(n-2)*c(n-1)-1)/(c(n-2)+c(n-1)+2). Then it appears that (apart from signs), a(n) is the denominator of c(n). - Jonas Holmvall, Jun 21 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Wikipedia, Fibonacci number
• Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,1).

Crossrefs

Cf. A000045, A130196 (denominator).

The a(2*n) appear in A179135. - Johannes W. Meijer, Jul 01 2010

Programs

GAP

a:=[0,1,1,1,3,5];; for n in [7..40] do a[n]:=4*a[n-3]+a[n-6]; od; a; # Muniru A Asiru, Oct 16 2018

Maple

nmax:=39; x(0):=0: x(1):=1/2:for n from 2 to nmax do x(n):=x(n-1)+x(n-2) od: for n from 0 to nmax do a(n):= numer(x(n)) od: seq(a(n),n=0..nmax); # Johannes W. Meijer, Jul 01 2010
with(combinat):f:=n->fibonacci(n):L:=n->f(n)+2*f(n-1):seq(numer(f(n)/L(n)), n=0..39); # Gary Detlefs, Dec 11 2010

Mathematica

f[n_]:=Numerator[Fibonacci[n]/Fibonacci[n+3]];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011*)
Numerator[LinearRecurrence[{1,1},{0,1/2},40]] (* Harvey P. Dale, Aug 08 2014 *)
CoefficientList[Series[-x (1 + x + x^2 - x^3 + x^4)/((x^2 + x - 1) (x^4 - x^3 + 2 x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 08 2014 *)
LinearRecurrence[{0, 0, 4, 0, 0, 1},{0, 1, 1, 1, 3, 5},40] (* Ray Chandler, Aug 03 2015 *)
a[n_]:=If[Mod[n,3]==0, Fibonacci[n]/2, Fibonacci[n]]; Array[a, 40, 0] (* Stefano Spezia, Oct 16 2018 *)

Formula

a(n) = (a(n-1)*A131534(n) + a(n-2)*A131534(n+2))/A131534(n+1) for n > 1.

a(3*n) = A001076(n) = (a(3*n-1) + a(3*n-2))/2;

a(3*n+1) = A033887(n) = 2*a(3*n-1) + a(3*n-2);

a(3*n+2) = A015448(n+1) = a(3*n-1) + 2*a(3*n-2).

From Johannes W. Meijer, Jul 01 2010: (Start)

a(2*n) = A001906(n)/A131534(n+1) for n >= 0 and a(2*n+1) = A179131(n)/5 for n >= 1.

a(6*n+2) - 2*a(6*n) = A134493(n);

2*a(6*n+1) - a(6*n+3) = A023039(n);

2*a(6*n+4) - a(6*n+2) = A134497(n);

a(6*n+5) - 2*a(6*n+3) = A103134(n);

2*a(6*n+4) - a(6*n+6) = A075796(n).

(End)

From Gary Detlefs, Dec 11 2010: (Start)

a(n) = numerator(A000045(n)/A000032(n)).

If n mod 3 = 0 then a(n) = Fibonacci(n)/2, else a(n)= Fibonacci(n). (End)

G.f.: -x*(1 + x + x^2 - x^3 + x^4) / ( (x^2 + x - 1)*(x^4 - x^3 + 2*x^2 + x + 1) ). - R. J. Mathar, Mar 08 2011

a(n) = 4*a(n-3) + a(n-6). - Muniru A Asiru, Oct 16 2018

Extensions

Typo in title corrected by Johannes W. Meijer, Jun 26 2010