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 A122737 #41 Mar 04 2025 08:36:09
%S A122737 0,2,6,20,72,274,1086,4438,18570,79174,342738,1502472,6656436,
%T A122737 29756910,134061570,608072340,2774495160,12726088630,58646299650,
%U A122737 271401086380,1260750482760,5876782098790,27479558368170,128861594138750,605869334122602,2855527261156394,13488568550452446
%N A122737 Expansion of 1 - 3*x - sqrt(1 - 6*x + 5*x^2).
%C A122737 Numbers of perifusenes with one internal vertex (see Cyvin et al. for precise definition).
%C A122737 For n>=2, a(n) is also the number of bi-wall directed polygons with perimeter 2n+2. Let us denote unit steps as follows: W=(-1,0), E=(1,0), N=(0,1), S=(0,-1). A bi-wall directed polygon is a self-avoiding polygon which can be factored as uv, where (1) u is a path which starts with an N step, ends with an S step, and can make N, E and S steps, and (2) v is a path which starts with a W step, ends with a W step, and can make W, S and E steps.
%H A122737 G. C. Greubel, <a href="/A122737/b122737.txt">Table of n, a(n) for n = 1..1000</a>
%H A122737 Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Ramirez/ramirez10.html">Counting prefixes of skew Dyck paths</a>, J. Int. Seq., Vol. 24 (2021), Article 21.8.2.
%H A122737 S. J. Cyvin, F. Zhang and J. Brunvoll, <a href="https://dx.doi.org/10.1007/BF01164209">Enumeration of perifusenes with one internal vertex: A complete mathematical solution</a>, J. Math. Chem., 11 (1992), 283-292.
%H A122737 Svjetlan Feretic, <a href="https://www.bib.irb.hr:8443/538510/"> Generating functions for bi-wall directed polygons</a>, in: Proc. of the Seventh Int. Conf. on Lattice Path Combinatorics and Applications (eds. S. Rinaldi and S. G. Mohanty), Siena, 2010, 147-151.
%H A122737 Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See p. 19.
%F A122737 For n>=1, a(n+1) = (3^(n+1)/(n*2^n))*Sum_{i=0..floor((n+1)/2)} ((-5/9)^i*binomial(n,i)*binomial(2*n-2*i,n-1)).
%F A122737 G.f.: 1/x - 3 - (1-x)/x/G(0), where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 24 2013
%F A122737 G.f.: (1-3*x - (1-5*x)*G(0))/x, where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 25 2013
%F A122737 a(n) ~ 5^(n-1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 29 2013
%F A122737 D-finite with recurrence: n*a(n) + 3*(-2*n+3)*a(n-1) + 5*(n-3)*a(n-2) = 0. - _R. J. Mathar_, Jan 23 2020
%F A122737 a(n) = 2*A002212(n-1), n>1. - _R. J. Mathar_, Jan 23 2020
%e A122737 There exist a(4)=20 bi-wall directed polygons with perimeter 2*4 + 2 = 10.
%t A122737 CoefficientList[Series[1 - 3*x - Sqrt[1 - 6*x + 5*x^2], {x,0,50}], x] (* _G. C. Greubel_, Mar 19 2017 *)
%o A122737 (PARI) x='x+O('x^66); concat([0],Vec(1-3*x-sqrt(1-6*x+5*x^2))) \\ _Joerg Arndt_, May 27 2013
%K A122737 nonn
%O A122737 1,2
%A A122737 _N. J. A. Sloane_, Sep 24 2006
%E A122737 Terms a(8)-a(20), better title, and extended edits from _Svjetlan Feretic_, May 24 2013