A293543 -id:A293543 - cp's OEIS frontend

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, 13029389, 21081995, 34111385, 55193380

Offset: 0

N. J. A. Sloane

			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 + ...

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).

Cf. A000045 (Fibonacci(n)).
Cf. A082115 (Fibonacci(n) (mod 3)).
Cf. A293543 (ceiling(Fibonacci(n)/3)).
Cf. A293544 (round(Fibonacci(n)/3)).

[Floor(Fibonacci(n)/3): n in [0..40]]; // Vincenzo Librandi, Jul 09 2012

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

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 *)

vector(40, n, n--; fibonacci(n)\3) \\ Altug Alkan, Nov 06 2015

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

[floor(fibonacci(n)/3) for n in (0..40)] # G. C. Greubel, May 22 2019

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

A293544 a(n) = round(Fibonacci(n)/3).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 7, 11, 18, 30, 48, 78, 126, 203, 329, 532, 861, 1394, 2255, 3649, 5904, 9552, 15456, 25008, 40464, 65473, 105937, 171410, 277347, 448756, 726103, 1174859, 1900962, 3075822, 4976784, 8052606, 13029390, 21081995, 34111385, 55193380

Offset: 0

Clark Kimberling, Oct 12 2017

a(n) is the integer k that minimizes | k/Fibonacci(n) - 1/3 |.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, -1, 1, 1).

Cf. A000045 (Fibonacci(n)).
Cf. A004696 (floor(Fibonacci(n)/3)).
Cf. A293543 (ceiling(Fibonacci(n)/3)).

Table[Round[Fibonacci[n]/3], {n, 0, 20}] (* Eric W. Weisstein, Feb 08 2025 *)
Round[Fibonacci[Range[0, 20]]/3] (* Eric W. Weisstein, Feb 08 2025 *)
LinearRecurrence[{1, 1, 0, -1, 1, 1}, {0, 0, 1, 1, 2, 3}, {0, 20}] (* Eric W. Weisstein, Feb 08 2025 *)
CoefficientList[Series[-(x^3/((-1 + x + x^2) (1 + x^4))), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 08 2025 *)
Table[(Fibonacci[n] + (-1)^n Sin[n Pi/4] (Cos[n Pi/2] + Sqrt[2] Sin[n Pi/2]))/3, {n, 0, 20}] (* Eric W. Weisstein, Feb 08 2025 *)

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

a(n) = a(n-1) + a(n-2) - a(n-4) + a(n-5) + a(n-6) for n >= 7.

cp's OEIS Frontend

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

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