A178959 - OEIS

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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

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OFFSET

0,1

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

LINKS

Table of n, a(n) for n=0..101.

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

FORMULA

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

EXAMPLE

0.652703644666139302296566746461370407999248645631861225527517243735868355...

MATHEMATICA

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

PROG

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