A128834 Periodic sequence 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, 0, -1, -1, 0, 1, 1

Offset

0,1

Comments

Unsigned version in A011655.

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 0, z = -1. - Michael Somos, Nov 27 2019

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).

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

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) = 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)

a(n) = -hypergeom([-n/2-1, -(n+1)/2], [-n-2], 4). - Peter Luschny, Dec 17 2016

a(3*n) = 0, a(6*n + 1) = a(6*n + 2) = 1, a(6*n + 4) = a(6*n + 5) = -1 for all n in Z. - Michael Somos, Nov 27 2019

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(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013
(Magma) I:=[0, 1]; [n le 2 select I[n] else Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

Crossrefs

Differs only by a shift from A010892.

Cf. A123331 (inverse Mobius transf.)

Sequence in context: A092220 A011655 A102283 * A022928 A000494 A022933

Adjacent sequences: A128831 A128832 A128833 * A128835 A128836 A128837

Keyword

sign,mult,easy

Author

Philippe Deléham, Apr 13 2007

Status

approved