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A163811 Expansion of (1 - x) * (1 - x^10) / ((1 - x^5) * (1 - x^6)) in powers of x.

%I A163811 #15 Aug 04 2017 20:04:41
%S A163811 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,
%T A163811 -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,
%U A163811 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,-1,0,0,0,1,0,-1,0
%N A163811 Expansion of (1 - x) * (1 - x^10) / ((1 - x^5) * (1 - x^6)) in powers of x.
%H A163811 G. C. Greubel, <a href="/A163811/b163811.txt">Table of n, a(n) for n = 0..1000</a>
%H A163811 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, -1, 0, -1).
%F A163811 Euler transform of length 10 sequence [ -1, 0, 0, 0, 1, 1, 0, 0, 0, -1].
%F A163811 a(2*n) = a(3*n) = 0 unless n=0, a(6*n + 1) = -1, a(6*n + 5) = a(0) = 1.
%F A163811 a(-n) = -a(n) unless n=0. a(n+6) = a(n) unless n=0 or n=-6.
%F A163811 G.f.: (1 - x + x^2 - x^3 + x^4) / (1 + x^2 + x^4).
%F A163811 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 A163811 1 - x + x^5 - x^7 + x^11 - x^13 + x^17 - x^19 + x^23 - x^25 + x^29 + ...
%t A163811 Join[{1}, LinearRecurrence[{0, -1, 0, -1}, {-1, 0, 0, 0}, 50]] (* _G. C. Greubel_, Aug 04 2017 *)
%o A163811 (PARI) {a(n) = (n==0) + [0, -1, 0, 0, 0, 1][n%6 + 1]}
%o A163811 (PARI) {a(n) = (n==0) - kronecker(-12, n)}
%Y A163811 A163817(n) = -a(n) unless n=0. A163817(n) = (-1)^n * a(n).
%Y A163811 Convolution inverse of A163812.
%K A163811 sign,easy
%O A163811 0,1
%A A163811 _Michael Somos_, Aug 04 2009, Aug 09 2009