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%I #64 Dec 14 2023 05:27:03
%S 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,
%T -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,
%U -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
%N Periodic sequence 0,1,1,0,-1,-1,...
%C Unsigned version in A011655.
%C 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
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1).
%H C. Kimberling, <a href="http://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%F a(n+1) = a(n) - a(n-1) for n>=1, with a(0)=0, a(1)=1.
%F G.f.: x * (1 + x) / (1 + x^3).
%F Euler transform of length 6 sequence [1, -1, -1, 0, 0, 1]. - _Michael Somos_, Apr 15 2007
%F 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
%F 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
%F a(n) = A010892(n-1). - _R. J. Mathar_, Feb 08 2008
%F a(n) = A010892(n+5). - _Jaume Oliver Lafont_, Dec 05 2008
%F 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
%F a(n) = 2*sin(n*Pi/3)/sqrt(3). - _Jaume Oliver Lafont_, Dec 05 2008
%F From _Wolfdieter Lang_, Jul 18 2010: (Start)
%F 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).
%F a(n) = S(n-1,1) = U(n-1,1/2) with S(-1,1)=0. (End)
%F a(n) = -hypergeom([-n/2-1, -(n+1)/2], [-n-2], 4). - _Peter Luschny_, Dec 17 2016
%F 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
%e G.f. = x + x^2 - x^4 - x^5 + x^7 + x^8 - x^10 - x^11 + x^13 + x^14 - x^16 + ...
%t PadRight[{},120,{0,1,1,0,-1,-1}] (* or *) LinearRecurrence[{1,-1}, {0,1}, 120] (* _Harvey P. Dale_, May 08 2014 *)
%t a[ n_] := (-1)^Quotient[n, 3] Sign[Mod[n, 3]]; (* _Michael Somos_, Apr 26 2015 *)
%t a[ n_] := {1, 1, 0, -1, -1, 0}[[Mod[n, 6, 1]]]; (* _Michael Somos_, Apr 26 2015 *)
%o (PARI) {a(n) = [0, 1, 1, 0, -1, -1][n%6 + 1]};
%o (Sage)
%o def A128834():
%o x, y = 0, -1
%o while True:
%o yield -x
%o x, y = y, -x + y
%o a = A128834(); [next(a) for i in range(40)] # _Peter Luschny_, Jul 11 2013
%o (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
%Y Differs only by a shift from A010892.
%Y Cf. A123331 (inverse Mobius transf.)
%K sign,mult,easy
%O 0,1
%A _Philippe Deléham_, Apr 13 2007
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