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%I #9 Jul 09 2019 13:53:18
%S 1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,
%T 0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,
%U -1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0,2,-1,0,1,-2,0
%N Expansion of x * (1 - x)^2 * (1 - x^2) / (1 - x^6) in powers of x.
%F Euler transform of length 6 sequence [ -2, -1, 0, 0, 0, 1].
%F Multiplicative with a(2^e) = 2*(-1)^e if e>0, a(3^e) = 0 if e>0, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).
%F G.f.: x * (1 - x)^2 / ((1 - x + x^2) * (1 + x + x^2)). - _Michael Somos_, May 04 2015
%F G.f.: -(f(x) + 3*f(-x)) / 2 where f(x) := x / (1 - x + x^2). - _Michael Somos_, May 04 2015
%F a(n) = -a(3 - n) = a(n+6), a(3*n) = 0 for all n in Z.
%e G.f. = x - 2*x^2 + 2*x^4 - x^5 + x^7 - 2*x^8 + 2*x^10 - x^11 + x^13 - 2*x^14 + ...
%t a[ n_] := {0, 1, -2, 0, 2, -1} [[ Mod[n, 6] + 1]]; (* _Michael Somos_, May 04 2015 *)
%t PadRight[{},120,{1,-2,0,2,-1,0}] (* _Harvey P. Dale_, Jul 09 2019 *)
%o (PARI) {a(n) = [0, 1, -2, 0, 2, -1][n%6 + 1]};
%K sign,easy,mult
%O 1,2
%A _Michael Somos_, Sep 02 2005
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