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%I #33 Apr 11 2023 02:06:17
%S 0,1,0,-1,1,1,-1,0,1,0,0,0,0,1,0,-1,1,1,-1,0,1,0,0,0,0,1,0,-1,1,1,-1,
%T 0,1,0,0,0,0,1,0,-1,1,1,-1,0,1,0,0,0,0,1,0,-1,1,1,-1,0,1,0,0,0,0,1,0,
%U -1,1,1,-1,0,1,0,0,0,0,1,0,-1,1,1,-1,0,1,0,0,0,0,1,0,-1,1,1,-1,0,1,0,0,0,0,1,0,-1
%N Expansion of 1/(1 + x^2 - x^3 - x^5).
%C Partial sums are A112689.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,1,0,1).
%F G.f.: 1/((1+x^2)*(1-x^3)).
%F a(n) = Sum_{k=0..n} Sum_{j=0..floor((k+1)/2)} (-1)^(k-j)*C(k-j+1, j-1).
%F a(n+12) = a(n) = 1/6 + A057077(n+1)/2 + A061347(n+1)/3. - _R. J. Mathar_, Feb 23 2009
%F a(n+10) = (A000100(n) mod 2)*(-1)^(1 + floor(n/2)). - _John M. Campbell_, Jul 07 2016
%F From _Ilya Gutkovskiy_, Jul 07 2016: (Start)
%F E.g.f.: (3*sin(x) + 3*cos(x) + exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/6.
%F a(n) = (3*sin(Pi*n/2) + 3*cos(Pi*n/2) - 4*cos(2*Pi*n/3) + 1)/6. (End)
%F a(n) = 2*floor(n/4) + floor((n+2)/3) - floor(n/3) - floor(n/2). - _Ridouane Oudra_, Mar 11 2023
%t LinearRecurrence[{0, -1, 1, 0, 1}, {0, 1, 0, -1, 1}, 100] (* _Vincenzo Librandi_, Jul 07 2016 *)
%o (PARI) concat(0, Vec(1/(1+x^2-x^3-x^5) + O(x^80))) \\ _Michel Marcus_, Jul 07 2016
%o (PARI) a(n) = round(real((exp(-2/3*I*n*Pi)*(-4+(3+3*I)*exp((I*n*Pi)/6) + 2*exp((2*I*n*Pi)/3) + (3-3*I)*exp((7*I*n*Pi)/6) - 4*exp((4*I*n*Pi)/3)))/12)) \\ _Colin Barker_, Jul 07 2016
%Y Cf. A000100, A057077, A061347, A112689.
%K easy,sign
%O 0,1
%A _Paul Barry_, Sep 15 2005
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