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

%I A004696 #59 Feb 09 2025 10:07:09
%S A004696 0,0,0,0,1,1,2,4,7,11,18,29,48,77,125,203,329,532,861,1393,2255,3648,
%T A004696 5903,9552,15456,25008,40464,65472,105937,171409,277346,448756,726103,
%U A004696 1174859,1900962,3075821,4976784,8052605,13029389,21081995,34111385,55193380
%N A004696 a(n) = floor(Fibonacci(n)/3).
%H A004696 Vincenzo Librandi, <a href="/A004696/b004696.txt">Table of n, a(n) for n = 0..1000</a>
%H A004696 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,1,-1,-1).
%F A004696 G.f.: x^4*(1 +x^3 +x^4) / ((1-x^8)*(1-x-x^2)).
%F A004696 a(n) = (A000045(n) - A082115(n))/3. - _R. J. Mathar_, Jul 14 2012
%F A004696 From _Vladimir Reshetnikov_, Nov 05 2015: (Start)
%F A004696 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.
%F A004696 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)
%F A004696 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
%e A004696 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 + ...
%p A004696 seq(iquo(fibonacci(n),3),n=0..40); # _Zerinvary Lajos_, Apr 20 2008
%t A004696 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 *)
%t A004696 Floor[Fibonacci[Range[0, 40]]/3] (* _G. C. Greubel_, May 22 2019 *)
%t A004696 Table[Floor[Fibonacci[n]/3], {n, 0, 20}] (* _Eric W. Weisstein_, Feb 09 2025 *)
%t A004696 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 *)
%o A004696 (Magma) [Floor(Fibonacci(n)/3): n in [0..40]]; // _Vincenzo Librandi_, Jul 09 2012
%o A004696 (PARI) vector(40, n, n--; fibonacci(n)\3) \\ _Altug Alkan_, Nov 06 2015
%o A004696 (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
%o A004696 (Sage) [floor(fibonacci(n)/3) for n in (0..40)] # _G. C. Greubel_, May 22 2019
%Y A004696 Cf. A000045 (Fibonacci(n)).
%Y A004696 Cf. A082115 (Fibonacci(n) (mod 3)).
%Y A004696 Cf. A293543 (ceiling(Fibonacci(n)/3)).
%Y A004696 Cf. A293544 (round(Fibonacci(n)/3)).
%K A004696 nonn,easy
%O A004696 0,7
%A A004696 _N. J. A. Sloane_