A152822 Periodic sequence [1,1,0,1] of length 4.

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

Offset

0,1

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for characteristic functions.

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

Formula

a(n) = 3/4 - (1/4)*(-1)^n + (1/2)*cos(n*Pi/2);

a(n+4) = a(n) with a(0) = a(1) = a(3) = 1 and a(2) = 0;

O.g.f.: (1+z+z^3)/(1-z^4);

From Paolo P. Lava, Dec 15 2008: (Start)

a(n) = (1/8)*((n mod 4) - ((n+1) mod 4) + 3*((n+2) mod 4)((n+3) mod 4)), with n >= 0.

a(n) = (1/4)*(3 + i^n + (-i)^n - (-1)^n), with n >= 0 and i = sqrt(-1). (End)

a(n) = ceiling(cos(Pi*n/4)^2). - Wesley Ivan Hurt, Jun 12 2013

From Antti Karttunen, May 03 2022: (Start)

Multiplicative with a(p^e) = 1 for odd primes, and a(2^e) = [e > 1]. (Here [ ] is the Iverson bracket, i.e., a(2^e) = 0 if e=1, and 1 if e>1).

a(n) = A166486(2+n).

(End)

Dirichlet g.f.: zeta(s)*(1 - 1/2^s + 1/4^s). - Amiram Eldar, Dec 27 2022

Maple

a:= n-> [1, 1, 0, 1][1+irem(n, 4)]:
seq(a(n), n=0..104);  # Alois P. Heinz, Sep 01 2021

Prog

(PARI) a(n)=n%4!=2 \\ Jaume Oliver Lafont, Mar 24 2009
(PARI) A152822(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], f[k, 2]>1, 1)); }; \\ (After multiplicative formula) - Antti Karttunen, May 03 2022
(Python)
def A152822(n): return (1, 1, 0, 1)[n&3] # Chai Wah Wu, Jan 10 2023

Crossrefs

Characteristic function of A042965.

Cf. A026052, A026064, A320111 (inverse Möbius transform).

Sequence A166486 shifted by two terms.

Sequence in context: A085369 A188082 A046980 * A118831 A118828 A105234

Adjacent sequences: A152819 A152820 A152821 * A152823 A152824 A152825

Keyword

easy,nonn,mult

Author

Richard Choulet, Dec 13 2008

Extensions

More terms from Philippe Deléham, Dec 21 2008

Keyword:mult added by Andrew Howroyd, Jul 27 2018

Status

approved