A247392 Decimal expansion of 'v', a parking constant associated with the asymptotic variance of the number of cars that can be parked in a given interval.
0, 3, 8, 1, 5, 6, 3, 9, 9, 1, 9, 0, 4, 2, 6, 5, 0, 5, 3, 2, 9, 1, 0, 4, 4, 9, 8, 2, 2, 5, 3
Offset: 0
Examples
0.0381563991904265053291044982253...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Rényi Parking Constant, p. 279.
Links
- Eric Weisstein's MathWorld, Rényi's Parking Constants
Crossrefs
Cf. A050996.
Programs
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Mathematica
digits = 30; beta[x_] := Exp[-2*(Gamma[0, x] + Log[x] + EulerGamma)]; m = NIntegrate[beta[x], {x, 0, Infinity}, WorkingPrecision -> digits+5]; alpha[x_?NumericQ] := m - NIntegrate[beta[t], {t, 0, x}, WorkingPrecision -> digits+5]; v = 4*NIntegrate[((1 - Exp[-x])*alpha[x])/(x*Exp[x]) - ((x + Exp[-x] - 1)*alpha[x]^2)/((beta[x]*x^2)* Exp[2*x]), {x, 0, Infinity}, WorkingPrecision -> digits+5] - m; Join[{0}, First[RealDigits[v, 10, digits]]]
Formula
beta(x) = exp(-2*(Gamma(0, x) + log(x) + EulerGamma)), where Gamma(0,x) is the incomplete Gamma function,
m = A050996 = Integral_{x=0..oo} beta(x) dx,
alpha(x) = m - Integral_{t=0..x} beta(t) dt,
v = 4*Integral_{x=0..oo} ((1 - exp(-x))*alpha(x))/(x*exp(x)) - ((x + exp(-x) - 1)*alpha(x)^2)/((beta(x)*x^2)* exp(2*x)) dx - m.