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%I A247392 #10 May 17 2025 08:08:05
%S A247392 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 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.
%D A247392 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Rényi Parking Constant, p. 279.
%H A247392 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/RenyisParkingConstants.html">Rényi's Parking Constants</a>
%F A247392 beta(x) = exp(-2*(Gamma(0, x) + log(x) + EulerGamma)), where Gamma(0,x) is the incomplete Gamma function,
%F A247392 m = A050996 = Integral_{x=0..oo} beta(x) dx,
%F A247392 alpha(x) = m - Integral_{t=0..x} beta(t) dt,
%F A247392 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.
%e A247392 0.0381563991904265053291044982253...
%t A247392 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 A247392 Cf. A050996.
%K A247392 nonn,cons,more
%O A247392 0,2
%A A247392 _Jean-François Alcover_, Sep 16 2014