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A281727 a(n) = (-1)^n * 2 if n = 3*k and n!=0, otherwise a(n) = (-1)^n.

%I A281727 #11 Mar 21 2017 11:17:07
%S A281727 1,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1,
%T A281727 -2,1,-1,2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,
%U A281727 2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1
%N A281727 a(n) = (-1)^n * 2 if n = 3*k and n!=0, otherwise a(n) = (-1)^n.
%H A281727 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-1).
%F A281727 Euler transform of length 6 sequence [-1, 1, -1, -1, 0, 1].
%F A281727 a(n) = -b(n) where b() is multiplicative with b(2^e) = -1 if e>0, b(3^e) = 2 if e>0, b(p^e) = 1 otherwise.
%F A281727 G.f.: 1 - x / (1 + x) - x^3 / (1 + x^3).
%F A281727 G.f.: (1 - x + x^2 - x^3) / (1 + x^3).
%F A281727 G.f.: (1 - x) * (1 - x^3) * (1 - x^4) / ((1 - x^2) * (1 - x^6)).
%F A281727 a(n) = a(-n) for all n in Z.
%F A281727 a(3*n) = A280560(n) for all n in Z.
%e A281727 G.f. = 1 - x + x^2 - 2*x^3 + x^4 - x^5 + 2*x^6 - x^7 + x^8 - 2*x^9 + ...
%t A281727 a[ n_] := {2, -1, 1, -2, 1, -1}[[Mod[n, 6] + 1]] - Boole[n == 0];
%t A281727 a[ n_] := (-1)^n If[ n != 0 && Divisible[n, 3], 2, 1];
%t A281727 a[ n_] := SeriesCoefficient[ (1 - x + x^2 - x^3) / (1 + x^3), {x, 0, Abs[n]}];
%o A281727 (PARI) {a(n) = (-1)^n * if(n && n%3==0, 2, 1)};
%o A281727 (PARI) {a(n) = [2, -1, 1, -2, 1, -1][n%6 + 1] - (n==0)};
%o A281727 (PARI) {a(n) = n=abs(n); polcoeff( (1 - x + x^2 - x^3) / (1 + x^3) + x * O(x^n), n)};
%Y A281727 Cf. A280560.
%K A281727 sign
%O A281727 0,4
%A A281727 _Michael Somos_, Jan 28 2017