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Expansion of 1/(1+x-2*x^3).
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%I #38 Sep 08 2022 08:45:08

%S 1,-1,1,1,-3,5,-3,-3,13,-19,13,13,-51,77,-51,-51,205,-307,205,205,

%T -819,1229,-819,-819,3277,-4915,3277,3277,-13107,19661,-13107,-13107,

%U 52429,-78643,52429,52429,-209715,314573,-209715,-209715,838861,-1258291,838861,838861,-3355443,5033165,-3355443

%N Expansion of 1/(1+x-2*x^3).

%H G. C. Greubel, <a href="/A077973/b077973.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 (-1,0,2).

%F a(n) = -a(n-1) +2*a(n-3). - _Paul Curtz_, Apr 23 2008

%F a(n) = (-1)^n * A077950(n) = (1/5) * (2*A134142(n) + 1). - _Ralf Stephan_, Aug 18 2013

%F G.f.: G(0)/(2*(1-x^2)), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 25 2013

%F a(n) = (-1)^n * Sum_{k=1..floor((n+2)/2)} binomial(n+2-2*k, k-1)*(-2)^(k-1). - _Taras Goy_, Sep 18 2019

%t LinearRecurrence[{-1,0,2}, {1,-1,1}, 50] (* or *) CoefficientList[Series[ 1/(1+x-2x^3), {x,0,50}], x] (* _Harvey P. Dale_, Jun 09 2017 *)

%o (PARI) Vec(1/(1+x-2*x^3)+O(x^50)) \\ _Charles R Greathouse IV_, Sep 23 2012

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1+x-2*x^3) )); // _G. C. Greubel_, Jun 24 2019

%o (Sage) (1/(1+x-2*x^3)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 24 2019

%o (GAP) a:=[1,-1,1];; for n in [4..50] do a[n]:=-a[n-1]+2*a[n-3]; od; a; # _G. C. Greubel_, Jun 24 2019

%Y Cf. A077950, A134142.

%K sign,easy

%O 0,5

%A _N. J. A. Sloane_, Nov 17 2002, Jun 17 2007