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%I #25 Sep 08 2022 08:44:39
%S 1,-1,-1,-1,-2,-2,-1,1,5,11,18,24,25,15,-13,-65,-142,-234,-313,-327,
%T -199,163,838,1840,3041,4079,4279,2639,-2042,-10802,-23841,-39519,
%U -53155,-55989,-34982,25544,139225,308895,513547,692655
%N Imaginary Rabbits: imaginary part of a(0)=i; a(1)=-i; a(n) = a(n-1) + i*a(n-2), with i = sqrt(-1).
%C Second differences give -a(n-2).
%D Charles Gely, "Lapins imaginaires et valeurs propres". Quadrature Quarterly #30.
%H Vincenzo Librandi, <a href="/A014291/b014291.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,-1).
%F a(n) = 2*a(n-1) - a(n-2) - a(n-4).
%F G.f.: (1-x)*(1-2*x)/(1-2*x+x^2+x^4).
%F a(n) = Im( -i*J(n, -i) + J(n-1, -i) ), where J(n,x) is the Jacobsthal polynomial, whose coefficient array is A128099, and i = sqrt(-1). - _G. C. Greubel_, Jun 15 2019
%t CoefficientList[Series[(2*x-1)*(x-1)/(x^4+x^2-2*x+1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 23 2012 *)
%o (Magma) I:=[1, -1, -1, -1]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) - Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Oct 23 2012
%o (PARI) my(x='x+O('x^50)); Vec((1-x)*(1-2*x)/(1-2*x+x^2+x^4)) \\ _G. C. Greubel_, Jun 13 2019
%o (Sage) ((1-x)*(1-2*x)/(1-2*x+x^2+x^4)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 13 2019
%o (GAP) a:=[1,-1,-1,-1];; for n in [5..50] do a[n]:=2*a[n-1]-a[n-2]-a[n-4]; od; a; # _G. C. Greubel_, Jun 13 2019
%Y Cf. A128099.
%K sign,easy
%O 0,5
%A _Olivier Gérard_
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