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A178959 Decimal expansion of the site percolation threshold for the (3,6,3,6) Kagome Archimedean lattice. 0
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, 2 (list; constant; graph; refs; listen; history; text; internal format)
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
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).
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
CROSSREFS
Sequence in context: A197265 A198107 A004554 * A370113 A266998 A021609
KEYWORD
nonn,cons,easy
AUTHOR
Jonathan Vos Post, Dec 22 2012
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)