This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A052947 #30 Jul 02 2025 16:01:58
%S A052947 1,0,1,2,1,4,5,6,13,16,25,42,57,92,141,206,325,488,737,1138,1713,2612,
%T A052947 3989,6038,9213,14016,21289,32442,49321,75020,114205,173662,264245,
%U A052947 402072,611569,930562,1415713,2153700,3276837,4985126,7584237,11538800
%N A052947 Expansion of 1/(1-x^2-2*x^3).
%C A052947 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 A052947 G. C. Greubel, <a href="/A052947/b052947.txt">Table of n, a(n) for n = 0..1000</a>
%H A052947 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1006">Encyclopedia of Combinatorial Structures 1006</a>
%H A052947 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2).
%F A052947 a(n) = a(n-2) + 2*a(n-3).
%F A052947 a(n) = Sum_{alpha=RootOf(-1+z^2+2*z^3)} (1/52)*(3 +17*alpha -2*alpha^2)*alpha^(-1-n).
%F A052947 a(n) = Sum_{k=0..floor(n/2)} C(k,n-2*k)*2^(n-2*k). - _Paul Barry_, Oct 16 2004
%F A052947 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 A052947 spec:= [S,{S=Sequence(Prod(Union(Prod(Union(Z,Z),Z),Z),Z))}, unlabeled]: seq(combstruct[count ](spec,size=n), n=0..20);
%p A052947 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 A052947 LinearRecurrence[{0,1,2}, {1,0,1}, 50] (* _G. C. Greubel_, Oct 21 2019 *)
%o A052947 (PARI) my(x='x+O('x^50)); Vec(1/(1-x^2-2*x^3)) \\ _G. C. Greubel_, Oct 21 2019
%o A052947 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x^2-2*x^3) )); // _G. C. Greubel_, Oct 21 2019
%o A052947 (Sage)
%o A052947 def A052947_list(prec):
%o A052947 P.<x> = PowerSeriesRing(ZZ, prec)
%o A052947 return P(1/(1-x^2-2*x^3)).list()
%o A052947 A052947_list(50) # _G. C. Greubel_, Oct 21 2019
%o A052947 (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 A052947 Column k=2 of A219946. - _Alois P. Heinz_, Dec 01 2012
%K A052947 easy,nonn
%O A052947 0,4
%A A052947 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052947 More terms from _James Sellers_, Jun 05 2000