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 A128834 Periodic sequence 0,1,1,0,-1,-1,... 26
 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 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) a(n) = -hypergeom([-n/2-1, -(n+1)/2], [-n-2], 4). - Peter Luschny, Dec 17 2016 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 (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

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Last modified December 14 16:59 EST 2018. Contains 318103 sequences. (Running on oeis4.)