A088911 Period 6: repeat [1, 1, 1, 0, 0, 0].

1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Oct 22 2003

Keywords

• nonn
• easy

Comments

For periodic sequences having a period of 2*k and composed of k ones followed by k zeros we have a(n) = floor(((n+k) mod 2*k)/k). Sequences of this form are A000035(n+1) (k=1), A133872(n) (k=2), this sequence (k=3), A131078(n) (k=4), and A112713(n-1) (k=5). - Gary Detlefs, May 17 2011

Links

• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Crossrefs

Cf. A000035, A010701, A079979, A133872, A131078, A112713.

Programs

Magma

&cat [[1, 1, 1, 0, 0, 0]^^30]; // Wesley Ivan Hurt, Jul 05 2016

Maple

seq(op([1, 1, 1, 0, 0, 0]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

Mathematica

CoefficientList[Series[(1 + x + x^2)/(1 - x^6), {x, 0, 50}], x]
Flatten[Table[{1,1,1,0,0,0},{20}]] (* Harvey P. Dale, Jul 17 2011 *)

PARI

a(n)=n%6<3 \\ Jaume Oliver Lafont, Mar 17 2009

Python

def A088911(n): return int(n % 6 < 3) # Chai Wah Wu, May 25 2022

Formula

G.f.: (1+x+x^2)/(1-x^6) = 1/((1-x)*(1+x)*(1-x+x^2)).

a(n) = a(n-6) for n>=6, a(0)=a(1)=a(2)=1, a(3)=a(4)=a(5)=0.

a(n) = ((-1)^floor((5*n + 2)/3) + 1)/2 = ( (-1)^floor(n/3) + 1 )/2. [Simplified by Bruno Berselli, Jul 09 2013]

a(n) = Sum_{k=0..floor(n/2)} U(n-2k, 1/2). - Paul Barry, Nov 15 2003

From Paul Barry, Mar 14 2004: (Start)

Partial sums of expansion of 1/(1+x^3), see A131531.

a(n) = 2*sin(Pi*n/3 + Pi/6)/3 + cos(Pi*n)/6 + 1/2. (End)

a(n) = floor(((n+3) mod 6)/3).

a(n) = floor((5*n-1)/3) mod 2. - Gary Detlefs, May 17 2011

a(n) = 1/2 + cos(Pi*n/3)/3 + sin(Pi*n/3)/sqrt(3) + (-1)^n/6. - R. J. Mathar, Oct 08 2011

a(n) = floor(((n+2)^2)/3) mod 2. - Wesley Ivan Hurt, Jun 29 2013

a(n) = A079979(n) + A079979(n-1) + A079979(n-2). - R. J. Mathar, Jul 10 2015

a(n) = a(n-1) - a(n-3) + a(n-4) for n > 3. - Wesley Ivan Hurt, Jul 05 2016

a(n) = 2*floor(n/6) - floor(n/3) + 1. - Ridouane Oudra, Dec 14 2021

E.g.f.: (2*cosh(x) + exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + sinh(x))/3. - Stefano Spezia, Aug 04 2025