A106664 - OEIS

%I #25 Mar 14 2024 13:55:01

%S -1,1,2,5,4,1,-8,-15,-16,1,32,65,64,1,-128,-255,-256,1,512,1025,1024,

%T 1,-2048,-4095,-4096,1,8192,16385,16384,1,-32768,-65535,-65536,1,

%U 131072,262145,262144,1,-524288,-1048575,-1048576,1,2097152,4194305,4194304,1,-8388608,-16777215,-16777216,1,33554432

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

%C Superseeker finds that a(n+2) - a(n) = A090131(n+1) (or with different signs, see A078069).

%C Floretion Algebra Multiplication Program, FAMP Code: 2ibaseiseq[ + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e]

%H G. C. Greubel, <a href="/A106664/b106664.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,-2,2).

%F a(n) = (1/2)*(A010673(n) - A099087(n+2)).

%F 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 From _G. C. Greubel_, Sep 08 2021: (Start)

%F a(n) = (1-(-1)^n)/2 - 2^((n+1)/2)*cos((n+1)*Pi/4).

%F a(n) = A000035(n) - A146559(n).

%F E.g.f.: sinh(x) - exp(x)*(cos(x) - sin(x)). (End)

%t 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 (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 (SageMath)

%o def A106664_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( sinh(x) -exp(x)*(cos(x)-sin(x)) ).egf_to_ogf().list()

%o A106664_list(50) # _G. C. Greubel_, Sep 08 2021

%Y Cf. A000035, A010673, A078069, A090131, A099087, A146559.

%K easy,sign

%O 0,3

%A _Creighton Dement_, May 13 2005