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%I #33 Sep 08 2022 08:45:47
%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
%N Expansion of (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^6)) in powers of x.
%H G. C. Greubel, <a href="/A163806/b163806.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 6 sequence [1, -1, -1, 0, 0, 1].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (2*v - 1) - (3*v - 2) * (2*u - 1).
%F a(n) is multiplicative with a(3^e) = 0^e, a(2^e) = -(-1)^e if e>0, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).
%F a(3*n) = 0 unless n=0, a(6*n + 4) = a(6*n + 5) = -1, a(6*n + 1) = a(6*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). - Corrected by _Bruno Berselli_, Apr 06 2011
%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 A128834(n) = a(n) unless n=0. A163810(n) = -a(n) unless n=0. A163804(n) = (-1)^n * a(n). Convolution inverse of A163805.
%e G.f. = 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},104]] (* _Ray Chandler_, Sep 15 2015 *)
%o (PARI) {a(n) = (n==0) + [0, 1, 1, 0, -1, -1][n%6 + 1]};
%o (PARI) {a(n) = (n==0) - (-1)^n * kronecker(-3, n)};
%o (Magma) I:=[1,1]; [1] cat [n le 2 select I[n] else Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018
%Y Cf. A128834, A163804, A163805, A163810.
%K sign,easy,mult
%O 0,1
%A _Michael Somos_, Aug 04 2009
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