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A163804 Expansion of (1 - x) * (1 - x^4) / ((1 - x^2) * (1 - x^3)) in powers of x.

%I A163804 #19 Jan 18 2025 09:35:15
%S A163804 1,-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 A163804 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 A163804 -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
%N A163804 Expansion of (1 - x) * (1 - x^4) / ((1 - x^2) * (1 - x^3)) in powers of x.
%H A163804 G. C. Greubel, <a href="/A163804/b163804.txt">Table of n, a(n) for n = 0..1000</a>
%H A163804 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1).
%F A163804 Euler transform of length 4 sequence [ -1, 1, 1, -1].
%F A163804 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 - v - u * (4 - 2*v - u).
%F A163804 a(3*n) = 0 unless n=0, a(3*n + 1) = -1, a(3*n + 2) = a(0) = 1.
%F A163804 a(-n) = -a(n) unless n=0. a(n+3) = a(n) unless n=0 or n=-3.
%F A163804 G.f.: (1 + x^2) / (1 + x + x^2).
%F A163804 G.f.: A(x) = 1 / (1 + x / (1 + x^2)) = 1 - x / (1 + x / (1 - x / (1 + x))). - _Michael Somos_, Jan 03 2013
%F A163804 a(n) = A057078(n-2), n>1. - _R. J. Mathar_, Aug 06 2009
%F A163804 From _Ridouane Oudra_, Jan 09 2025: (Start)
%F A163804 a(n) = 3*floor((n+1)/3) - n, for n>0.
%F A163804 a(n) = 2*sin(4*Pi*n/3)/sqrt(3), for n>0.
%F A163804 a(n) = - A102283(n), for n>0.
%F A163804 a(n) = - A106510(n), for n>0. (End)
%e A163804 1 - x + x^2 - x^4 + x^5 - x^7 + x^8 - x^10 + x^11 - x^13 + x^14 + ...
%p A163804 1, seq(2*sin(4*Pi*n/3)/sqrt(3), n=1..100); # _Ridouane Oudra_, Jan 09 2025
%t A163804 Join[{1},LinearRecurrence[{-1, -1},{-1, 1},105]] (* _Ray Chandler_, Sep 15 2015 *)
%o A163804 (PARI) {a(n) = (n==0) + [0, -1, 1][n%3 + 1]}
%o A163804 (PARI) {a(n) = (n==0) - kronecker(-3, n)}
%Y A163804 A106510(n) = -a(n) unless n=0. Convolution inverse of A117659.
%Y A163804 Cf. A102283.
%K A163804 sign,easy
%O A163804 0,1
%A A163804 _Michael Somos_, Aug 04 2009