A247392 - OEIS

%I #5 Sep 16 2014 03:50:10

%S 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

%N 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.

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

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/RenyisParkingConstants.html">Rényi's Parking Constants</a>

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

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

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

%F 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.

%e 0.0381563991904265053291044982253...

%t 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]]]

%Y Cf. A050996.

%K nonn,cons,more

%O 0,2

%A _Jean-François Alcover_, Sep 16 2014