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A163817 Expansion of (1 - x^2) * (1 - x^5) / ((1 - x) * (1 - x^6)) in powers of x.

%I A163817 #17 Aug 04 2017 20:04:57
%S A163817 1,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,
%T A163817 0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,
%U A163817 0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0
%N A163817 Expansion of (1 - x^2) * (1 - x^5) / ((1 - x) * (1 - x^6)) in powers of x.
%H A163817 G. C. Greubel, <a href="/A163817/b163817.txt">Table of n, a(n) for n = 0..1000</a>
%H A163817 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, -1, 0, -1).
%F A163817 Euler transform of length 6 sequence [ 1, -1, 0, 0, -1, 1].
%F A163817 a(n) is multiplicative with a(2^e) = a(3^e) = 0^e, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).
%F A163817 a(2*n) = a(3*n) = 0 unless n=0, a(6*n + 5) = -1, a(6*n + 1) = a(0) = 1.
%F A163817 a(-n) = -a(n) unless n=0. a(n+6) = a(n) unless n=0 or n=-6.
%F A163817 G.f.: (1 + x + x^2 + x^3 + x^4) / (1 + x^2 + x^4).
%F A163817 a(n) = A134667(n), n>0. - _R. J. Mathar_, Aug 05 2009
%F A163817 G.f. A(x) = 1 + x / (1 + x^4 / (1 + x^2)) = 1 / (1 - x / (1 + x / (1 - x^3 / (1 + x^2 / (1 - x / (1 + x)))))) . - _Michael Somos_, Jan 03 2013
%e A163817 1 + x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...
%t A163817 Join[{1}, LinearRecurrence[{0, -1, 0, -1}, {1, 0, 0, 0}, 50]] (* _G. C. Greubel_, Aug 04 2017 *)
%o A163817 (PARI) {a(n) = (n==0) + [0, 1, 0, 0, 0, -1][n%6 + 1]}
%o A163817 (PARI) {a(n) = (n==0) + kronecker(-12, n)}
%Y A163817 A163811(n) = -a(n) unless n=0. A163811(n) = (-1)^n * a(n).
%Y A163817 Convolution inverse of A163818.
%K A163817 sign,easy,mult
%O A163817 0,1
%A A163817 _Michael Somos_, Aug 04 2009