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%I #33 Dec 15 2022 12:46:50
%S 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,
%T 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,
%U 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
%N Decimal expansion of the site percolation threshold for the (3,6,3,6) Kagome Archimedean lattice.
%C 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.
%C 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
%H M. F. Sykes and J. W. Essam, <a href="https://doi.org/10.1063/1.1704215">Exact critical percolation probabilities for site and bond problems in two dimensions</a>, J. Math. Phys. 5, 1117 (1964).
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Percolation_threshold">Percolation threshold</a>
%F Equals 1 - 2*sin(Pi/18) = 1 = 1 - 2*cos(4*Pi/9) = 1 - A130880.
%e 0.652703644666139302296566746461370407999248645631861225527517243735868355...
%t RealDigits[1 - 2 Sin[Pi/18], 10, 105][[1]] (* _Alonso del Arte_, Dec 22 2012 *)
%o (PARI) 1-2*sin(Pi/18) \\ _Charles R Greathouse IV_, Jan 03 2013
%Y Cf. A130880, A174849, A332437, A332438.
%K nonn,cons,easy
%O 0,1
%A _Jonathan Vos Post_, Dec 22 2012
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