OFFSET
0,4
COMMENTS
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1006
Index entries for linear recurrences with constant coefficients, signature (0,1,2).
FORMULA
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
MAPLE
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
MATHEMATICA
LinearRecurrence[{0, 1, 2}, {1, 0, 1}, 50] (* G. C. Greubel, Oct 21 2019 *)
PROG
(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)
def A052947_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/(1-x^2-2*x^3)).list()
A052947_list(50) # G. C. Greubel, Oct 21 2019
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 05 2000
STATUS
approved