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A128834 Periodic sequence 0,1,1,0,-1,-1,... 23
0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Unsigned version in A011655.

LINKS

Table of n, a(n) for n=0..104.

Index entries for sequences related to Chebyshev polynomials.

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

FORMULA

a(n+1) = a(n) - a(n-1) for n>=1, with a(0)=0, a(1)=1.

G.f.: x * (1 + x) / (1 + x^3).

Euler transform of length 6 sequence [ 1, -1, -1, 0, 0, 1]. - Michael Somos, Apr 15 2007

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 + 2*u*v - 2*u^2*v. - Michael Somos, Apr 15 2007

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = v - u^3 + 3*u*v - 3*u^3*v. - Michael Somos, Apr 15 2007

a(n) = (1/6)*(-(n mod 6)+((n+2) mod 6)+((n+3) mod 6)-((n+5) mod 6)), with n>=0. - Paolo P. Lava, Jun 11 2007

a(n) = A010892(n-1). - R. J. Mathar, Feb 08 2008

a(n) = A010892(n+5). - Jaume Oliver Lafont, Dec 05 2008

a(n) is multiplicative with a(3^e) = 0^e, a(p^e) = 1 if p == 1 (mod 3), a(p^e) = (-1)^e if p == 2 (mod 3). - Michael Somos, Apr 15 2007

a(n) = 2*sin(n*Pi/3)/sqrt(3). - Jaume Oliver Lafont, Dec 05 2008

From Wolfdieter Lang, Jul 18 2010: (Start)

O.g.f.: x/(1-x+x^2) = x*S(x), with S(x) o.g.f. for Chebyshev S(n,1) = U(n,1/2) = A010892(n).

a(n) = S(n-1,1) = U(n-1,1/2) with S(-1,1)=0. (End)

EXAMPLE

G.f. = x + x^2 - x^4 - x^5 + x^7 + x^8 - x^10 - x^11 + x^13 + x^14 - x^16 + ...

MATHEMATICA

PadRight[{}, 120, {0, 1, 1, 0, -1, -1}] (* or *) LinearRecurrence[{1, -1}, {0, 1}, 120] (* Harvey P. Dale, May 08 2014 *)

a[ n_] := (-1)^Quotient[n, 3] Sign[Mod[n, 3]]; (* Michael Somos, Apr 26 2015 *)

a[ n_] := {1, 1, 0, -1, -1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Apr 26 2015 *)

PROG

(PARI) {a(n) = [0, 1, 1, 0, -1, -1][n%6 + 1]};

(Sage)

def A128834():

    x, y = 0, -1

    while true:

        yield -x

        x, y = y, -x + y

a = A128834(); [a.next() for i in range(40)]  # Peter Luschny, Jul 11 2013

CROSSREFS

Differs only by a shift from A010892.

Sequence in context: A092220 A011655 A102283 * A022928 A000494 A022933

Adjacent sequences:  A128831 A128832 A128833 * A128835 A128836 A128837

KEYWORD

sign,mult,easy,changed

AUTHOR

Philippe Deléham, Apr 13 2007

STATUS

approved

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Last modified December 3 12:53 EST 2016. Contains 278738 sequences.