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)).
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 *)
Table[Floor[Fibonacci[n]/3], {n, 0, 20}] (* Eric W. Weisstein, Feb 09 2025 *)
Table[(8 Fibonacci[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, {n, 0, 20}] (* Eric W. Weisstein, Feb 09 2025 *)
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
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved