A267409 Decimal expansion of the constant describing the average number of edges of a random labeled planar graph with n vertices.

2, 2, 1, 3, 2, 6, 5, 2, 3, 8, 5, 7, 4, 4, 2, 1, 7, 8, 7, 6, 1, 6, 7, 4, 9, 0, 4, 7, 6, 3, 1, 9, 5, 2, 6, 6, 3, 8, 6, 5, 1, 9, 5, 6, 2, 5, 1, 1, 5, 5, 4, 2, 1, 5, 9, 2, 7, 9, 7, 1, 8, 2, 7, 1, 7, 7, 1, 9, 5, 9, 7, 6, 4, 8, 7, 0, 3, 8, 8, 5, 0, 8, 3, 2

Offset

1,1

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..51004

Omer Gimenez, Marc Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309-329.

Formula

Equals lim E[Xn]/n, where Xn is the number of edges of a random labeled planar graph with n vertices.

Equals Km(A266389), where function t->Km(t) is defined in the PARI code.

Example

2.21326523857442...

Prog

(PARI)
A266389= 0.6263716633;
Y1(t)  = t^2 * (1-t) * (18 + 36*t + 5*t^2);
Y2(t)  = 2 * (3+t) * (1+2*t) * (1+3*t)^2;
Y(t)   = (1+2*t) / ((1+3*t)*(1-t)) * exp(-Y1(t)/Y2(t)) - 1;
A1(t)  = log(1+t) * (3*t-1) * (1+t)^3 / (16*t^3);
A2(t)  = log(1+2*t) * (1+3*t) * (1-t)^3 / (32*t^3);
A3(t)  = (1-t) * (185*t^4 + 698*t^3 - 217*t^2 - 160*t + 6);
A4(t)  = 64*t * (1+3*t)^2 * (3+t);
A(t)   = A1(t) + A2(t) + A3(t) / A4(t);
R(t)   = 1/16 * sqrt(1+3*t) * (1/t - 1)^3 * exp(A(t));
Km(t)  = -R'(t)/(R(t)*Y'(t));
Km(A266389)

Crossrefs

Cf. A266389, A266390, A267410, A267412.

Sequence in context: A296786 A145141 A103360 * A104469 A372437 A144112

Adjacent sequences: A267406 A267407 A267408 * A267410 A267411 A267412

Keyword

nonn,cons

Author

Gheorghe Coserea, Jan 13 2016

Status

approved