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

0,2

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Rényi Parking Constant, p. 279.

Links

Table of n, a(n) for n=0..30.

Eric Weisstein's MathWorld, Rényi's Parking Constants

Formula

beta(x) = exp(-2*(Gamma(0, x) + log(x) + EulerGamma)), where Gamma(0,x) is the incomplete Gamma function,

m = A050996 = integral_{0..infinity} beta(x) dx,

alpha(x) = m - integral_{0..x} beta(t) dt,

v = 4*integral_{0..infinity} (((1 - exp(-x))*alpha(x))/(x*exp(x)) - ((x + exp(-x) - 1)*alpha(x)^2)/((beta(x)*x^2)* exp(2*x)) dx.

Example

0.0381563991904265053291044982253...

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]]]

Crossrefs

Cf. A050996.

Sequence in context: A238169 A341414 A086245 * A219995 A021266 A054399

Adjacent sequences: A247389 A247390 A247391 * A247393 A247394 A247395

Keyword

nonn,cons,more

Author

Jean-François Alcover, Sep 16 2014

Status

approved