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 A176006 #41 Jul 06 2025 23:45:38
%S A176006 1,2,4,10,32,122,516,2322,10880,52466,258564,1296282,6589728,33887466,
%T A176006 175966212,921353250,4858956288,25786112994,137604139012,737922992938,
%U A176006 3974647310112,21493266631002,116642921832964,635074797251890
%N A176006 The number of branching configurations of RNA (see Sankoff, 1985) with n or fewer hairpins.
%C A176006 a(n) is the number of dissections of a convex (n+2)-sided polygon by non-intersecting diagonals such that selected least two consecutive sides of the polygon will be in the same sub-polygon. - _Muhammed Sefa Saydam_, Jul 02 2025
%H A176006 Vincenzo Librandi, <a href="/A176006/b176006.txt">Table of n, a(n) for n = 0..200</a>
%H A176006 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A176006 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A176006 David Sankoff, <a href="https://doi.org/10.1137/0145048">Simultaneous solution of the RNA folding, alignment and protosequence problems</a>, SIAM J. Appl. Math 45(5) (1985), 810-825.
%H A176006 David Sankoff, <a href="https://pdfs.semanticscholar.org/7ce8/d1231c8c00ddc36de23aaf4cf1225a130f3e.pdf?_ga=2.152919417.1954741913.1595406974-1181019897.1595406974">Simultaneous solution of the RNA folding, alignment and protosequence problems</a>, SIAM J. Appl. Math 45(5) (1985), 810-825.
%F A176006 G.f.: (3 - x - sqrt(1 - 6*x + x^2))/(2*(1 - x)).
%F A176006 Conjecture : n*a(n) +(9-7*n)*a(n-1) +(7*n-12)*a(n-2) +(3-n)*a(n-3)=0. - _R. J. Mathar_, Jul 24 2012
%F A176006 a(n) ~ 2^(1/4)*(3 + 2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 20 2012
%F A176006 a(n) = Sum_{x+y=n+1} A006318(x), for y >= 2, x >= -1 and A006318(-1) = 1. - _Muhammed Sefa Saydam_, Jul 02 2025
%e A176006 For n = 3, the a(3) = 10 branching configurations with 3 or fewer hairpins are: unfolded, (), ()(), (()()), ()()(), (()())(), ()(()()), (()()()), ((()())()), and (()(()())).
%t A176006 CoefficientList[Series[(3-x-Sqrt[1-6*x+x^2])/(2*(1-x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A176006 (PARI) my(x='x+O('x^50)); Vec((3-x-sqrt(1-6*x+x^2))/(2*(1-x))) \\ _G. C. Greubel_, Mar 22 2017
%Y A176006 The cumulative sums of A155069.
%K A176006 easy,nonn
%O A176006 0,2
%A A176006 _Lee A. Newberg_, Apr 05 2010