A122737 Expansion of 1 - 3*x - sqrt(1 - 6*x + 5*x^2).

0, 2, 6, 20, 72, 274, 1086, 4438, 18570, 79174, 342738, 1502472, 6656436, 29756910, 134061570, 608072340, 2774495160, 12726088630, 58646299650, 271401086380, 1260750482760, 5876782098790, 27479558368170, 128861594138750, 605869334122602, 2855527261156394, 13488568550452446

Offset: 1

Author

N. J. A. Sloane, Sep 24 2006

Keywords

• nonn

Comments

Numbers of perifusenes with one internal vertex (see Cyvin et al. for precise definition).

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.

Examples

			There exist a(4)=20 bi-wall directed polygons with perimeter 2*4 + 2 = 10.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, Counting prefixes of skew Dyck paths, J. Int. Seq., Vol. 24 (2021), Article 21.8.2.
• S. J. Cyvin, F. Zhang and J. Brunvoll, Enumeration of perifusenes with one internal vertex: A complete mathematical solution, J. Math. Chem., 11 (1992), 283-292.
• Svjetlan Feretic, Generating functions for bi-wall directed polygons, in: Proc. of the Seventh Int. Conf. on Lattice Path Combinatorics and Applications (eds. S. Rinaldi and S. G. Mohanty), Siena, 2010, 147-151.
• Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See p. 19.

Programs

Mathematica

CoefficientList[Series[1 - 3*x - Sqrt[1 - 6*x + 5*x^2], {x,0,50}], x] (* G. C. Greubel, Mar 19 2017 *)

PARI

x='x+O('x^66); concat([0],Vec(1-3*x-sqrt(1-6*x+5*x^2))) \\ Joerg Arndt, May 27 2013

Formula

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)).

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

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

a(n) ~ 5^(n-1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013

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

a(n) = 2*A002212(n-1), n>1. - R. J. Mathar, Jan 23 2020

Extensions

Terms a(8)-a(20), better title, and extended edits from Svjetlan Feretic, May 24 2013