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%I #17 Sep 08 2022 08:45:08
%S 1,-1,-1,0,2,1,-2,-3,1,5,2,-6,-7,4,13,3,-17,-16,14,33,2,-47,-35,45,82,
%T -10,-127,-72,137,199,-65,-336,-134,401,470,-267,-871,-203,1138,1074,
%U -935,-2212,-139,3147,2351,-3008,-5498,657,8506,4841,-9163,-13347,4322,22510,9025,-26832,-31535,17807,58367
%N Expansion of (1-x)/(1+x^2+x^3).
%H G. C. Greubel, <a href="/A078032/b078032.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,-1).
%F a(0)=1, a(1)=-1, a(2)=-1, a(n) = -a(n-2) - a(n-3). - _Harvey P. Dale_, Sep 14 2012
%F a(n) = (-1)^n * A097333(-2-n) for all n in Z. - _Michael Somos_, Nov 02 2021
%e G.f. = 1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - 3*x^7 + x^8 + 5*x^9 + ... - _Michael Somos_, Nov 02 2021
%p seq(coeff(series((1-x)/(1+x^2+x^3), x, n+1), x, n), n = 0..60); # _G. C. Greubel_, Aug 05 2019
%t CoefficientList[Series[(1-x)/(1+x^2+x^3),{x,0,60}],x] (* or *) LinearRecurrence[{0,-1,-1},{1,-1,-1},60] (* _Harvey P. Dale_, Sep 14 2012 *)
%t a[ n_] := If[n >= 0, SeriesCoefficient[(1 - x)/(1 + x^2 + x^3), {x, 0, n}], SeriesCoefficient[(x^2 - 1)/(1 + x + x^3), {x, 0, 1 - n}]]; (* _Michael Somos_, Nov 02 2021 *)
%o (PARI) my(x='x+O('x^60)); Vec((1-x)/(1+x^2+x^3)) \\ _G. C. Greubel_, Aug 05 2019
%o (PARI) {a(n) = if(n>=0, polcoeff( (1 - x)/(1 + x^2 + x^3) + x*O(x^n), n), polcoeff( (x^2 - 1)/(1 + x + x^3) + x*O(x^(1-n)), 1-n))}; /* _Michael Somos_, Nov 02 2021 */
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x)/(1+x^2+x^3) )); // _G. C. Greubel_, Aug 05 2019
%o (Sage) ((1-x)/(1+x^2+x^3)).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 05 2019
%o (GAP) a:=[1,-1,-1];; for n in [4..60] do a[n]:=-a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Aug 05 2019
%Y Cf. A097333.
%K sign
%O 0,5
%A _N. J. A. Sloane_, Nov 17 2002
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