A281727 - OEIS

%I #11 Mar 21 2017 11:17:07

%S 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 -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 2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1,-2,1,-1,2,-1,1

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-1).

%F Euler transform of length 6 sequence [-1, 1, -1, -1, 0, 1].

%F 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 G.f.: 1 - x / (1 + x) - x^3 / (1 + x^3).

%F G.f.: (1 - x + x^2 - x^3) / (1 + x^3).

%F G.f.: (1 - x) * (1 - x^3) * (1 - x^4) / ((1 - x^2) * (1 - x^6)).

%F a(n) = a(-n) for all n in Z.

%F a(3*n) = A280560(n) for all n in Z.

%e G.f. = 1 - x + x^2 - 2*x^3 + x^4 - x^5 + 2*x^6 - x^7 + x^8 - 2*x^9 + ...

%t a[ n_] := {2, -1, 1, -2, 1, -1}[[Mod[n, 6] + 1]] - Boole[n == 0];

%t a[ n_] := (-1)^n If[ n != 0 && Divisible[n, 3], 2, 1];

%t a[ n_] := SeriesCoefficient[ (1 - x + x^2 - x^3) / (1 + x^3), {x, 0, Abs[n]}];

%o (PARI) {a(n) = (-1)^n * if(n && n%3==0, 2, 1)};

%o (PARI) {a(n) = [2, -1, 1, -2, 1, -1][n%6 + 1] - (n==0)};

%o (PARI) {a(n) = n=abs(n); polcoeff( (1 - x + x^2 - x^3) / (1 + x^3) + x * O(x^n), n)};

%Y Cf. A280560.

%K sign

%O 0,4

%A _Michael Somos_, Jan 28 2017