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A163805 Expansion of (1 - x) * (1 - x^6) / ((1 - x^3) * (1 - x^4)) in powers of x.

%I A163805 #28 Nov 19 2024 00:47:51
%S A163805 1,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,
%T A163805 0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,
%U A163805 -1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0
%N A163805 Expansion of (1 - x) * (1 - x^6) / ((1 - x^3) * (1 - x^4)) in powers of x.
%H A163805 G. C. Greubel, <a href="/A163805/b163805.txt">Table of n, a(n) for n = 0..1000</a>
%H A163805 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1).
%F A163805 Euler transform of length 6 sequence [ -1, 0, 1, 1, 0, -1].
%F A163805 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v) - u * (2 - u) * (3 - 2*v).
%F A163805 a(2*n) = 0 unless n=0, a(4*n + 1) = -1, a(4*n + 3) = a(0) = 1.
%F A163805 a(-n) = -a(n) unless n=0. a(n+4) = a(n) unless n=0 or n=-4.
%F A163805 a(n) = - A117569(n) unless n=0. a(n) = (-1)^n * A117569(n).
%F A163805 Convolution inverse of A163806.
%F A163805 G.f.: (1 - x + x^2) / (1 + x^2).
%F A163805 G.f. A(x) = 1 - x / (1 + x^2) = 1 / (1 + x / (1 - x / (1 + x / (1 - x)))). - _Michael Somos_, Jan 03 2013
%F A163805 a(n) = A101455(n-2) = A056594(n-3), n>2. - _R. J. Mathar_, Aug 06 2009
%F A163805 E.g.f.: 1 - sin(x). - _Stefano Spezia_, Nov 16 2024
%F A163805 a(n) = sin(3*n*Pi/2), for n>0. - _Ridouane Oudra_, Nov 18 2024
%e A163805 G.f. = 1 - x + x^3 - x^5 + x^7 - x^9 + x^11 - x^13 + x^15 - x^17 + x^19 + ...
%p A163805 1, seq(sin(3*n*Pi/2), n=1..100); # _Ridouane Oudra_, Nov 18 2024
%t A163805 a[ n_] := Boole[n == 0] + {-1, 0, 1, 0}[[Mod[n, 4, 1]]]; (* _Michael Somos_, Sep 06 2015 *)
%o A163805 (PARI) {a(n) = (n==0) + [0, -1, 0, 1][n%4 + 1]};
%o A163805 (PARI) {a(n) = (n==0) - kronecker(-4, n)};
%Y A163805 Cf. A117569, A163806.
%K A163805 sign,easy
%O A163805 0,1
%A A163805 _Michael Somos_, Aug 04 2009