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A052947
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Expansion of 1/(1-x^2-2*x^3).
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20
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1, 0, 1, 2, 1, 4, 5, 6, 13, 16, 25, 42, 57, 92, 141, 206, 325, 488, 737, 1138, 1713, 2612, 3989, 6038, 9213, 14016, 21289, 32442, 49321, 75020, 114205, 173662, 264245, 402072, 611569, 930562, 1415713, 2153700, 3276837, 4985126, 7584237, 11538800
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = a(n-2) + 2*a(n-3).
a(n) = Sum_{alpha=RootOf(-1+z^2+2*z^3)} (1/52)*(3 +17*alpha -2*alpha^2)*alpha^(-1-n).
a(n) = Sum_{k=0..floor(n/2)} C(k,n-2*k)*2^(n-2*k). - Paul Barry, Oct 16 2004
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
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MAPLE
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spec:= [S, {S=Sequence(Prod(Union(Prod(Union(Z, Z), Z), Z), Z))}, unlabeled]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(coeff(series(1/(1-x^2-2*x^3), x, n+1), x, n), n = 0..50); # G. C. Greubel, Oct 21 2019
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MATHEMATICA
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LinearRecurrence[{0, 1, 2}, {1, 0, 1}, 50] (* G. C. Greubel, Oct 21 2019 *)
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PROG
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(PARI) my(x='x+O('x^50)); Vec(1/(1-x^2-2*x^3)) \\ G. C. Greubel, Oct 21 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x^2-2*x^3) )); // G. C. Greubel, Oct 21 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/(1-x^2-2*x^3)).list()
(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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