A004696 a(n) = floor(Fibonacci(n)/3).

0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 18, 29, 48, 77, 125, 203, 329, 532, 861, 1393, 2255, 3648, 5903, 9552, 15456, 25008, 40464, 65472, 105937, 171409, 277346, 448756, 726103, 1174859, 1900962, 3075821, 4976784, 8052605

Offset

0,7

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,0,0,1,-1,-1).

Formula

G.f.: x^4*(1 +x^3 +x^4) / ((1-x^8)*(1-x-x^2)).

a(n) = (A000045(n) - A082115(n))/3. - R. J. Mathar, Jul 14 2012

From Vladimir Reshetnikov, Nov 05 2015: (Start)

a(n) = (8*A000045(n) + 3*(-1)^n - 9 + cos(Pi*n/2)*(6 - 4*sin(Pi*n/4)) + 4*sqrt(2)*sin(Pi*n/4)*sin(Pi*n/2))/24.

E.g.f.: (cos(x)-cosh(x)-2*sinh(x))/4 + (sqrt(2)*cos(x/sqrt(2))+sin(x/sqrt(2))) *sinh(x/sqrt(2))/6 + 2*exp(x/2)*sinh(x*sqrt(5)/2))/( 3*sqrt(5)). (End)

The sequence b(n) = a(n+2) - a(n+1) - a(n) has period 8 and always 0 or 1. - Michael Somos, Nov 06 2015

Example

G.f. = x^4 + x^5 + 2*x^6 + 4*x^7 + 7*x^8 + 11*x^9 + 18*x^10 + 29*x^11 + 48*x^12 + ...

Maple

seq(iquo(fibonacci(n), 3), n=0..40); # Zerinvary Lajos, Apr 20 2008

Mathematica

CoefficientList[Series[x^4(x^4+x^3+1)/((1-x^8)(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 09 2012 *)
Floor[Fibonacci[Range[0, 40]]/3] (* G. C. Greubel, May 22 2019 *)

Prog

(Magma) [Floor(Fibonacci(n)/3): n in [0..40]]; // Vincenzo Librandi, Jul 09 2012
(PARI) vector(40, n, n--; fibonacci(n)\3) \\ Altug Alkan, Nov 06 2015
(PARI) concat(vector(4), Vec(x^4*(x^4+x^3+1)/((1-x^8)*(1-x-x^2)) + O(x^40))) \\ Altug Alkan, Nov 06 2015
(Sage) [floor(fibonacci(n)/3) for n in (0..40)] # G. C. Greubel, May 22 2019

Crossrefs

Sequence in context: A054352 A091838 A288219 * A293418 A289077 A228560

Adjacent sequences: A004693 A004694 A004695 * A004697 A004698 A004699

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved