A246686 Decimal expansion of 'mu', a percolation constant associated with the asymptotic threshold for 3-dimensional bootstrap percolation.

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Sep 01 2014

Keywords

• nonn
• cons
• easy

Examples

			0.4039127202987558379321142074495349887102719293775432644...
		

References

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

Links

• J. Balogh, B. Bollobás and R. Morris, Bootstrap percolation in three dimensions. arXiv:0806.4485 [math.CO], 2008-2009.
• Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024; p. 47.
• Eric Weisstein's MathWorld, Bootstrap Percolation

Crossrefs

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

Programs

Mathematica

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

Formula

Equals -Integral_{0..oo} log(1/2 - exp(-2*x)/2 + (1/2)*sqrt(1 + exp(-4*x) - 4*exp(-3*x) + 2*exp(-2*x))) dx.