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%I #14 Aug 04 2017 19:56:54
%S 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 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,-1,1,0
%N Expansion of (1 - x) * (1 - x^4) / ((1 - x^2) * (1 - x^3)) in powers of x.
%H G. C. Greubel, <a href="/A163804/b163804.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1, -1).
%F Euler transform of length 4 sequence [ -1, 1, 1, -1].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 - v - u * (4 - 2*v - u).
%F a(3*n) = 0 unless n=0, a(3*n + 1) = -1, a(3*n + 2) = a(0) = 1.
%F a(-n) = -a(n) unless n=0. a(n+3) = a(n) unless n=0 or n=-3.
%F G.f.: (1 + x^2) / (1 + x + x^2).
%F G.f. A(x) = 1 / (1 + x / (1 + x^2)) = 1 - x / (1 + x / (1 - x / (1 + x))). - _Michael Somos_, Jan 03 2013
%F a(n) = A057078(n-2), n>1. - _R. J. Mathar_, Aug 06 2009
%e 1 - x + x^2 - x^4 + x^5 - x^7 + x^8 - x^10 + x^11 - x^13 + x^14 + ...
%t Join[{1},LinearRecurrence[{-1, -1},{-1, 1},105]] (* _Ray Chandler_, Sep 15 2015 *)
%o (PARI) {a(n) = (n==0) + [0, -1, 1][n%3 + 1]}
%o (PARI) {a(n) = (n==0) - kronecker(-3, n)}
%Y A106510(n) = -a(n) unless n=0. Convolution inverse of A117659.
%K sign,easy
%O 0,1
%A _Michael Somos_, Aug 04 2009
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