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 A052919 #51 Jul 02 2025 16:01:58
%S A052919 2,3,7,19,55,163,487,1459,4375,13123,39367,118099,354295,1062883,
%T A052919 3188647,9565939,28697815,86093443,258280327,774840979,2324522935,
%U A052919 6973568803,20920706407,62762119219,188286357655,564859072963
%N A052919 a(n) = 1 + 2*3^(n-1) with a(0)=2.
%C A052919 It appears that if s(n) is a first order rational sequence of the form s(1)=3, s(n) = (2*s(n-1)+1)/(s(n-1)+2), n > 1, then s(n) = a(n)/(a(n)-2).
%C A052919 The binomial transform is 2, 5, 15, 51, 187, ...A007581 without the leading term. - _R. J. Mathar_, Apr 07 2022
%H A052919 Vincenzo Librandi, <a href="/A052919/b052919.txt">Table of n, a(n) for n = 0..1000</a>
%H A052919 Gennady Eremin, <a href="https://arxiv.org/abs/2012.12675">Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind</a>, arXiv:2012.12675 [math.CO], 2020.
%H A052919 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=902">Encyclopedia of Combinatorial Structures 902</a>
%H A052919 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A052919 a(n) = 1 + 2*3^(n-1) for n > 0 with a(0) = 2.
%F A052919 G.f.: (2 - 5*x + x^2)/((1-x)*(1-3*x)).
%F A052919 a(n) = 4*a(n-1) - 3*a(n-2), with a(0)=2, a(1)=3, a(2)=7.
%F A052919 a(0) = 2 and a(n) = A100702(n) for n >= 1. - _Omar E. Pol_, Mar 02 2012
%F A052919 a(n) = A104743(n) - A104743(n-1). - _J. M. Bergot_, Jun 07 2013
%p A052919 spec := [S,{S=Union(Sequence(Prod(Sequence(Z),Union(Z,Z))),Sequence(Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
%p A052919 seq(`if`(n=0, 2, 1 + 2*3^(n-1)), n=0..30); # _G. C. Greubel_, Oct 16 2019
%t A052919 Join[{2},Table[2*(3^n+1)-1,{n,0,30}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 14 2011*)
%t A052919 CoefficientList[Series[(2-5*x+x^2)/((1-x)*(1-3*x)),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 22 2012 *)
%t A052919 LinearRecurrence[{4,-3},{2,3,7},30] (* _Harvey P. Dale_, Dec 12 2017 *)
%o A052919 (Magma) I:=[2, 3, 7]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 22 2012
%o A052919 (PARI) vector(31, n, if(n==1, 2, 1+ 2*3^(n-2))) \\ _G. C. Greubel_, Oct 16 2019
%o A052919 (Sage) [2]+[1+2*3^(n-1) for n in (1..30)] # _G. C. Greubel_, Oct 16 2019
%o A052919 (GAP) Concatenation([2], List([1..30], n-> 1 + 2*3^(n-1) )); # _G. C. Greubel_, Oct 16 2019
%K A052919 easy,nonn
%O A052919 0,1
%A A052919 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052919 More terms from _James Sellers_, Jun 05 2000