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A106664 Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).

%I A106664 #25 Mar 14 2024 13:55:01
%S A106664 -1,1,2,5,4,1,-8,-15,-16,1,32,65,64,1,-128,-255,-256,1,512,1025,1024,
%T A106664 1,-2048,-4095,-4096,1,8192,16385,16384,1,-32768,-65535,-65536,1,
%U A106664 131072,262145,262144,1,-524288,-1048575,-1048576,1,2097152,4194305,4194304,1,-8388608,-16777215,-16777216,1,33554432
%N A106664 Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).
%C A106664 Superseeker finds that a(n+2) - a(n) = A090131(n+1) (or with different signs, see A078069).
%C A106664 Floretion Algebra Multiplication Program, FAMP Code: 2ibaseiseq[ + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e]
%H A106664 G. C. Greubel, <a href="/A106664/b106664.txt">Table of n, a(n) for n = 0..1000</a>
%H A106664 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,-2,2).
%F A106664 a(n) = (1/2)*(A010673(n) - A099087(n+2)).
%F A106664 a(n) = (1/2)*(1 - (-1)^n - (1-i)^(n+1) - (1+i)^(n+1)), with i=sqrt(-1). - _Ralf Stephan_, Nov 16 2010
%F A106664 From _G. C. Greubel_, Sep 08 2021: (Start)
%F A106664 a(n) = (1-(-1)^n)/2 - 2^((n+1)/2)*cos((n+1)*Pi/4).
%F A106664 a(n) = A000035(n) - A146559(n).
%F A106664 E.g.f.: sinh(x) - exp(x)*(cos(x) - sin(x)). (End)
%t A106664 CoefficientList[Series[(1-3x+x^2)/((1-x)(1+x)(1-2x+2x^2)),{x,0,60}],x] (* _Harvey P. Dale_, Mar 20 2013 *)
%o A106664 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!(  (1-3*x+x^2)/((1-x^2)*(1-2*x+2*x^2)) )); // _G. C. Greubel_, Sep 08 2021
%o A106664 (SageMath)
%o A106664 def A106664_list(prec):
%o A106664     P.<x> = PowerSeriesRing(QQ, prec)
%o A106664     return P( sinh(x) -exp(x)*(cos(x)-sin(x)) ).egf_to_ogf().list()
%o A106664 A106664_list(50) # _G. C. Greubel_, Sep 08 2021
%Y A106664 Cf. A000035, A010673, A078069, A090131, A099087, A146559.
%K A106664 easy,sign
%O A106664 0,3
%A A106664 _Creighton Dement_, May 13 2005