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 A052997 #32 Jul 02 2025 16:01:59
%S A052997 1,3,7,14,29,58,117,234,469,938,1877,3754,7509,15018,30037,60074,
%T A052997 120149,240298,480597,961194,1922389,3844778,7689557,15379114,
%U A052997 30758229,61516458,123032917,246065834,492131669,984263338,1968526677
%N A052997 Expansion of (1+x-x^3)/((1-2*x)*(1-x^2)).
%H A052997 Harvey P. Dale, <a href="/A052997/b052997.txt">Table of n, a(n) for n = 0..1000</a>
%H A052997 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1075">Encyclopedia of Combinatorial Structures 1075</a>
%H A052997 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F A052997 G.f.: -(-x+x^3-1)/(-1+x^2)/(-1+2*x).
%F A052997 Recurrence: {a(0)=1, -2*a(n)-a(n+1)+a(n+2)-1, a(1)= 3, a(2)=7, a(3)=14}, 11/6*2^n + Sum(-1/6*(2 + _alpha)*_alpha^(-1-n), _alpha=RootOf(-1 + _Z^2))
%F A052997 a(n) = 2*a(n-1)+1 for even n, otherwise a(n) = 2*a(n-1), with a(0)=1, a(1)=3. [_Bruno Berselli_, Jun 19 2014]
%F A052997 3*a(n) = 11*2^(n-1)-A000034(n) for n>0. - _R. J. Mathar_, Feb 27 2019
%p A052997 spec := [S,{S=Prod(Union(Sequence(Prod(Z,Z)),Z),Sequence(Union(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t A052997 f[s_List] := Block[{a = s[[-1]]}, Append[s, If[ OddQ@ Length@ s, 2a +1, 2a]]]; Join[{1}, Nest[f, {3}, 30]] (* or *)
%t A052997 CoefficientList[ Series[(1 + x - x^3)/(1 - 2x - x^2 + 2x^3), {x, 0, 30}], x] (* _Robert G. Wilson v_, Jul 20 2017 *)
%t A052997 LinearRecurrence[{2,1,-2},{1,3,7,14},40] (* _Harvey P. Dale_, May 27 2019 *)
%K A052997 nonn,easy
%O A052997 0,2
%A A052997 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052997 More terms from _James Sellers_, Jun 06 2000