A246686 - OEIS

%I #10 Jan 17 2020 05:56:39

%S 4,0,3,9,1,2,7,2,0,2,9,8,7,5,5,8,3,7,9,3,2,1,1,4,2,0,7,4,4,9,5,3,4,9,

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

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

%N Decimal expansion of 'mu', a percolation constant associated with the asymptotic threshold for 3-dimensional bootstrap percolation.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.18 Percolation Cluster Density Constants, pp. 371-378.

%H J. Balogh, B. Bollobás and R. Morris, <a href="http://arxiv.org/pdf/0806.4485.pdf">Bootstrap percolation in three dimensions.</a> arXiv:0806.4485v2 [math.CO] 31 Aug 2009

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants.</a> p. 47.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BootstrapPercolation.html">Bootstrap Percolation</a>

%F -integral_{0..infinity} log(1/2 - exp(-2*x)/2 + (1/2)*sqrt(1 + exp(-4*x) - 4*exp(-3*x) + 2*exp(-2*x))) dx.

%e 0.4039127202987558379321142074495349887102719293775432644...

%t mu = -NIntegrate[Log[1/2 - Exp[-2*x]/2 + (1/2)*Sqrt[1 + Exp[-4*x] - 4*Exp[-3*x] + 2 *Exp[-2*x]]] , {x, 0, Infinity}, WorkingPrecision -> 101]; RealDigits[mu] // First

%Y Cf. A086463 (analog 2-dimensional percolation constant).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Sep 01 2014