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A052947 Expansion of 1/(1-x^2-2*x^3). 16


%S 1,0,1,2,1,4,5,6,13,16,25,42,57,92,141,206,325,488,737,1138,1713,2612,

%T 3989,6038,9213,14016,21289,32442,49321,75020,114205,173662,264245,

%U 402072,611569,930562,1415713,2153700,3276837,4985126,7584237,11538800

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

%C a(n) is the number of compositions (ordered partitions) of n into parts 1 of one kind and part 2 of two kinds. - _Joerg Arndt_, Oct 22 2019

%H G. C. Greubel, <a href="/A052947/b052947.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1006">Encyclopedia of Combinatorial Structures 1006</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2).

%F a(n) = a(n-2) + 2*a(n-3).

%F a(n) = Sum_{alpha=RootOf(-1+z^2+2*z^3)} (1/52)*(3 +17*alpha -2*alpha^2)*alpha^(-1-n).

%F a(n) = Sum_{k=0..floor(n/2)} C(k,n-2*k)*2^(n-2*k). - _Paul Barry_, Oct 16 2004

%F If p[1]=0, p[2]=1, p[3]=2, p[i]=0, (i>3), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det A. - _Milan Janjic_, May 02 2010

%p spec:= [S,{S=Sequence(Prod(Union(Prod(Union(Z,Z),Z),Z),Z))}, unlabeled]: seq(combstruct[count ](spec,size=n), n=0..20);

%p seq(coeff(series(1/(1-x^2-2*x^3), x, n+1), x, n), n = 0..50); # _G. C. Greubel_, Oct 21 2019

%t LinearRecurrence[{0,1,2}, {1,0,1}, 50] (* _G. C. Greubel_, Oct 21 2019 *)

%o (PARI) my(x='x+O('x^50)); Vec(1/(1-x^2-2*x^3)) \\ _G. C. Greubel_, Oct 21 2019

%o (MAGMA) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x^2-2*x^3) )); // _G. C. Greubel_, Oct 21 2019

%o (Sage)

%o def A052947_list(prec):

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

%o return P(1/(1-x^2-2*x^3)).list()

%o A052947_list(50) # _G. C. Greubel_, Oct 21 2019

%o (GAP) a:=[1,0,1];; for n in [4..50] do a[n]:=a[n-2]+2*a[n-3]; od; a; # _G. C. Greubel_, Oct 21 2019

%Y Column k=2 of A219946. - _Alois P. Heinz_, Dec 01 2012

%K easy,nonn

%O 0,4

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _James A. Sellers_, Jun 05 2000

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Last modified December 13 06:26 EST 2019. Contains 329968 sequences. (Running on oeis4.)