A245672 Decimal expansion of k_3 = 3/(2*Pi*m_3), a constant associated with the asymptotic expansion of the probability that a three-dimensional random walk reaches a given point for the first time, where m_3 is A086231 (Watson's integral).
3, 1, 4, 8, 7, 0, 2, 3, 1, 3, 5, 9, 6, 2, 0, 1, 7, 8, 0, 7, 5, 1, 7, 3, 9, 1, 9, 4, 1, 8, 8, 0, 6, 8, 7, 7, 0, 5, 8, 9, 6, 3, 4, 2, 4, 5, 9, 0, 1, 4, 0, 5, 5, 1, 0, 8, 4, 0, 8, 0, 3, 0, 7, 2, 7, 3, 1, 0, 8, 0, 5, 9, 4, 7, 6, 1, 4, 6, 7, 3, 1, 9, 7, 9, 7, 5, 2, 0, 2, 4, 1, 2, 0, 2, 0, 4, 9, 6, 4, 0, 4, 2, 3, 4, 4
Offset: 0
Examples
0.314870231359620178075173919418806877058963424590140551084080307273108...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's Random Walk Constants, p. 324.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Eric Weisstein's MathWorld, Polya's Random Walk Constants
Crossrefs
Cf. A086231.
Programs
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Mathematica
k3 = 8*Sqrt[6]*Pi^2/(Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]); RealDigits[k3, 10, 105] // First
Formula
k_3 = 8*sqrt(6)*Pi^2/(Gamma(5/24)*Gamma(7/24)*Gamma(11/24)), where 'Gamma' is the Euler gamma function.
Asymptotic probability ~ k_3 / ||l||, where the norm ||l|| of the position of the lattice point l tends to infinity.