A232991 Period 6: repeat [1, 0, 0, 0, 1, 0].

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

Offset: 0

Author

N. J. A. Sloane, Dec 13 2013

Keywords

• nonn
• easy

References

• Andrews, George E., q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra. CBMS Regional Conference Series in Mathematics, 66. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. xii+130 pp. ISBN: 0-8218-0716-1 MR0858826 (88b:11063). See p. 105.

Links

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

Crossrefs

Cf. A089128, A232990.

Programs

Haskell

a232991 = (0 ^) . subtract 1 . gcd 6 . (+ 1)
a232991_list = cycle [1,0,0,0,1,0]
-- Reinhard Zumkeller, Apr 06 2015

Magma

/* By definition: */ &cat [[1,0,0,0,1,0]: n in [0..20]]; // Bruno Berselli, Feb 18 2015

Magma

[(((n+3) mod 6) mod 5) mod 2: n in [0..100]]; // Vincenzo Librandi, Feb 18 2015

Maple

A232991:=n->ceil((n+4)/6) - floor((n+4)/6) - (n mod 2): seq(A232991(n), n=0..100); # Wesley Ivan Hurt, Mar 13 2014

Mathematica

Table[Ceiling[(n + 4)/6] - Floor[(n + 4)/6] - Mod[n, 2], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 13 2014 *)
Table[Cos[Pi*n/2]/3 * (Cos[Pi*n/6] + 2*Cos[Pi*n/2] + Sqrt[3]*Sin[Pi*n/6]), {n, 0, 100}] (* Vaclav Kotesovec, Mar 23 2014 *)

PARI

a(n)=(n+9)\6 - (n+4)\6 - n%2 \\ Charles R Greathouse IV, Jul 17 2016

Python

def A232991(n): return int(not (n+1) % 6 & 3 ^ 1) # Chai Wah Wu, May 25 2022

Formula

a(n) = ceiling((n + 4)/6) - floor((n + 4)/6) - (n mod 2). - Wesley Ivan Hurt, Mar 13 2014

a(n) = cos(Pi*n/2)/3*(cos(Pi*n/6) + 2*cos(Pi*n/2) + sqrt(3)*sin(Pi*n/6)). - Vaclav Kotesovec, Mar 23 2014

G.f.: (1 + x^4)/(1 - x^6). - Bruno Berselli, Feb 18 2015

a(n) = if gcd(n+1, 6) > 1 then 0, otherwise 1. - Reinhard Zumkeller, Apr 06 2015

a(n) = a(n-6) for n > 5. - Wesley Ivan Hurt, Jun 20 2016

E.g.f.: (2*cosh(x) - sqrt(3)*sin(sqrt(3)*x/2)*sinh(x/2) + cos(sqrt(3)*x/2)*cosh(x/2))/3. - Ilya Gutkovskiy, Jun 21 2016

a(n) = gcd(gcd(floor((n+2)/3), 2), n) - 1. - Lechoslaw Ratajczak, Jul 30 2021

a(n) = sign((n+1) mod ((5+(-1)^n)/2)). - Wesley Ivan Hurt, Feb 04 2022