A259833 Decimal expansion of m_3, the expected number of returns to the origin in a three-dimensional random walk restricted to the region x >= y >= z.
1, 0, 6, 9, 3, 4, 1, 1, 2, 0, 6, 0, 6, 8, 8, 6, 6, 8, 2, 8, 2, 7, 7, 5, 7, 1, 6, 6, 8, 5, 9, 5, 5, 9, 2, 2, 9, 7, 8, 9, 9, 6, 5, 0, 2, 5, 8, 3, 5, 1, 7, 0, 7, 1, 5, 0, 8, 6, 7, 5, 4, 5, 9, 1, 4, 8, 4, 6, 2, 7, 1, 8, 9, 0, 4, 4, 5, 5, 9, 8, 5, 2, 7, 5, 4, 5, 2, 2, 3, 5, 8, 8, 7, 7, 5, 9, 4, 7, 6, 2, 2, 9, 8, 5, 3
Offset: 1
Examples
m_3 = 1.069341120606886682827757166859559229789965025835170715... Return probability is p_3 = 1 - 1/m_3 = 0.064844715377...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 326.
Links
- Eric Weisstein's MathWorld, Polya's Random Walk Constants
- J. Wimp and D. Zeilberger, How likely is Polya's drunkard to stay in x >= y >= z ? J. Statistical Physics 57, 1129-1135 (1989).
Programs
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Maple
evalf(Sum((2*n)!*hypergeom([1/2, -n-1, -n], [2, 2], 4)/(n!*(n+1)!*6^(2*n)), n=0..infinity), 120); # Vaclav Kotesovec, May 14 2016
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Mathematica
Sum[CatalanNumber[n]*HypergeometricPFQ[{1/2, -n - 1, -n}, {2, 2}, 4]/ 6^(2*n), {n, 0, 2*10^4}] // N // RealDigits // First (* Jul 06 2015, updated May 14 2016 *)
Formula
Sum_{n>=0} CatalanNumber(n) * 3F2(1/2,-n-1,-n; 2,2; 4) / 6^(2n), where 3F2 is the hypergeometric function.
Extensions
More terms from Vaclav Kotesovec, May 14 2016