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%I #32 Sep 08 2022 08:45:00
%S 1,2,7,24,85,302,1075,3828,13633,48554,172927,615888,2193517,7812326,
%T 27824011,99096684,352938073,1257007586,4476898903,15944711880,
%U 56787933445,202253224094,720335539171,2565513065700,9137210275441,32542656957722,115902391424047
%N Expansion of ( 1-2*x ) / ( (x-1)*(2*x^2+3*x-1) ).
%H Vincenzo Librandi, <a href="/A052986/b052986.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1060">Encyclopedia of Combinatorial Structures 1060</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-2).
%F G.f.: (1-2*x)/(1-4*x+x^2+2*x^3).
%F Recurrence: {a(0)=1, a(1)=2, -2*a(n)-3*a(n+1)+a(n+2)+1=0}.
%F a(n) = Sum(-1/136*(-13-27*r+6*r^2)*r^(-1-n) where r=RootOf(1-4*_Z+_Z^2+2*_Z^3)).
%F a(n) = (1/4+(2^(-3-n)*((3-sqrt(17))^n*(-5+3*sqrt(17))+(3+sqrt(17))^n*(5+3*sqrt(17))))/sqrt(17)). - _Colin Barker_, Sep 02 2016
%F 4*a(n) = 1+3*A007482(n)-2*A007482(n-1) - _R. J. Mathar_, Feb 27 2019
%p spec := [S,{S=Sequence(Union(Prod(Union(Sequence(Union(Z,Z)),Z),Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t Join[{a=1,b=2},Table[c=3*b+2*a-1;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011*)
%t LinearRecurrence[{4,-1,-2},{1,2,7},40] (* _Vincenzo Librandi_, Jun 23 2012 *)
%o (Magma) I:=[1, 2, 7]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 23 2012
%o (PARI) a(n) = round((1/4+(2^(-3-n)*((3-sqrt(17))^n*(-5+3*sqrt(17))+(3+sqrt(17))^n*(5+3*sqrt(17))))/sqrt(17))) \\ _Colin Barker_, Sep 02 2016
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 06 2000
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