A178959 Decimal expansion of the site percolation threshold for the (3,6,3,6) Kagome Archimedean lattice.

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

Offset: 0

Author

Jonathan Vos Post, Dec 22 2012

Keywords

• nonn
• cons
• easy

Comments

Consider an infinite graph where vertices are selected with probability p. The site percolation threshold is a unique value p_c such that if p > p_c an infinite connected component of selected vertices will almost surely exist, and if p < p_c an infinite connected component will almost surely not exist. This sequence gives p_c for the (3,6,3,6) Kagome Archimedean lattice.

This is one of the three real roots of x^3 - 3x^2 + 1. The other roots are 1 + A332437 = 2.879385241... and -(A332438 - 3) = - 0.5320888862... . - Wolfdieter Lang, Dec 13 2022

Examples

			0.652703644666139302296566746461370407999248645631861225527517243735868355...
		

Links

• M. F. Sykes and J. W. Essam, Exact critical percolation probabilities for site and bond problems in two dimensions, J. Math. Phys. 5, 1117 (1964).
• Wikipedia, Percolation threshold

Crossrefs

Cf. A130880, A174849, A332437, A332438.

Programs

Mathematica

RealDigits[1 - 2 Sin[Pi/18], 10, 105][[1]] (* Alonso del Arte, Dec 22 2012 *)

PARI

1-2*sin(Pi/18) \\ Charles R Greathouse IV, Jan 03 2013

Formula

Equals 1 - 2*sin(Pi/18) = 1 = 1 - 2*cos(4*Pi/9) = 1 - A130880.

Extensions

a(98) corrected and more terms from Georg Fischer, Jun 06 2024