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A166486 Periodic sequence [0,1,1,1] of length 4. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Sum_{k>0} a(k)/k/3^k = log(5)/4.

From Reinhard Zumkeller, Nov 30 2009: (Start)

a(n) = 1-A121262(n); characteristic function of numbers that are not multiples of 4; a(A042968(n))=1; a(A008586(n))=0;

A033436(n) = SUM(a(k)*(n-k): 0<=k<=n). (End)

A190621(n) = n * a(n).

LINKS

Table of n, a(n) for n=0..100.

M. Somos, Rational Function Multiplicative Coefficients

Index entries for characteristic functions

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

FORMULA

G.f.: (x + x^2 + x^3) / (1 - x^4) = x * (1 + x + x^2) / ((1 - x) * (1 + x) * (1 + x^2)) = x * (1 - x^3) / ((1 - x) * (1 - x^4)).

a(n) = (3 - i^n - (-i)^n - (-1)^n) / 4, where i=sqrt(-1).

Multiplicative with a(p^e) = (if p=2 then 0^(e-1) else 1), p prime and e>0. - Reinhard Zumkeller, Nov 30 2009

a(n) = 1/2*((n^3+n) mod 4). - Gary Detlefs, Mar 20 2010

a(n) = (3*(n mod 4)+(n+1 mod 4)+(n+2 mod 4)-(n+3 mod 4))/8 (cf. forms of modular arithmetic of Paolo P. Lava, i.e., see A146094). - Bruno Berselli, Sep 27 2010

a(n) = (Fibonacci(n)*Fibonacci(3n) mod 3)/2. - Gary Detlefs Dec 21 2010

Euler transform of length 4 sequence [ 1, 0, -1, 1]. - Michael Somos, Feb 12 2011

Dirichlet g.f. (1-1/4^s)*zeta(s). - R. J. Mathar, Feb 19 2011

a(n) = Fibonacci(n)^2 mod 3. - Gary Detlefs, May 16 2011

a(n) = -1/4*cos(Pi*n)-1/2*cos(1/2*Pi*n)+3/4. - Leonid Bedratyuk, May 13, 2012

For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

a(n) = ceiling(n/4) - floor(n/4). - Wesley Ivan Hurt, Jun 20 2014

a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015

EXAMPLE

G.f. = x + x^2 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^13 + x^14 + ...

MAPLE

seq(1/2*((n^3+n) mod 4), n=0..50); # Gary Detlefs, Mar 20 2010

MATHEMATICA

PadRight[{}, 120, {0, 1, 1, 1}] (* Harvey P. Dale, Jul 04 2013 *)

Table[Ceiling[n/4] - Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)

a[ n_] := Sign[ Mod[n, 4]]; (* Michael Somos, May 05 2015 *)

PROG

(PARI) {a(n) = !!(n%4)};

(MAGMA) [Ceiling(n/4)-Floor(n/4) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

CROSSREFS

Cf. A016628, A152822, A164985, A165132. First difference of A057353.

Cf. A168185, A145568, A168184, A168182, A168181, A109720, A097325, A011558, A011655, A000035.

Sequence in context: A140318 A060584 A098725 * A046978 A075553 A131729

Adjacent sequences:  A166483 A166484 A166485 * A166487 A166488 A166489

KEYWORD

nonn,mult,easy

AUTHOR

Jaume Oliver Lafont, Oct 15 2009

STATUS

approved

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Last modified November 19 04:01 EST 2017. Contains 294912 sequences.