A050996 -id:A050996 - cp's OEIS frontend

A243266 Decimal expansion of a parking constant related to the asymptotic expected number of cars, assuming random car lengths.

9, 8, 4, 8, 7, 1, 2, 8, 2, 5, 2, 5, 9, 9, 5, 3, 0, 4, 4, 7, 2, 7, 9, 5, 2, 2, 1, 5, 0, 7, 0, 5, 9, 5, 3, 2, 3, 1, 2, 7, 6, 0, 9, 1, 7, 0, 4, 1, 0, 3, 7, 4, 9, 5, 8, 1, 3, 6, 5, 2, 3, 2, 5, 5, 2, 0, 6, 5, 3, 7, 9, 3, 8, 8, 4, 0, 7, 3, 1, 6, 0, 6, 4, 3, 1, 8, 7, 0, 0, 9, 7, 4, 9, 4, 6, 3, 0, 0, 6, 7

Offset: 0

Jean-François Alcover, Jun 02 2014

			0.9848712825259953044727952215...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Renyi's parking constant, p. 279.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's MathWorld, Rényi's Parking Constants

Cf. A050996.

(1-1/2^((Sqrt[17]-1)/4))*Sqrt[Pi]*Gamma[Sqrt[17]/2]/(Gamma[(Sqrt[17]+1)/4]*Gamma[(Sqrt[17]+3)/4]^2) // RealDigits[#, 10, 100]& // First

(1-1/2^((sqrt(17)-1)/4))*sqrt(Pi)*gamma(sqrt(17)/2)/(gamma((sqrt(17)+1)/4)*gamma((sqrt(17)+3)/4)^2) \\ G. C. Greubel, Feb 14 2017

(1-1/2^((sqrt(17)-1)/4))*sqrt(Pi)*GAMMA(sqrt(17)/2)/(GAMMA((sqrt(17)+1)/4)*GAMMA((sqrt(17)+3)/4)^2), where GAMMA is the Euler Gamma function.

A050995 Reduced denominators of series expansion for integrand in Renyi's parking constant.

Original entry on oeis.org

1, 1, 2, 9, 144, 1800, 64800, 39690, 12700800, 171460800, 17146080000, 518668920000, 2390026383360, 3606379096320000, 449813829104640000, 139161153379248000000, 4453156908135936000000

Offset: 0

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Renyi's Parking Constants.

Cf. A050994, A050996.

nmax = 20; Denominator[CoefficientList[Series[Exp[-2*EulerGamma+2*ExpIntegralEi[-x]]/x^2, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 12 2019 *)

A242943 Decimal expansion of the mean car density associated with Solomon's variation in Renyi's one-dimensional parking problem.

Original entry on oeis.org

8, 0, 8, 6, 5, 2, 5, 1, 8, 3, 5, 0, 2, 1, 2, 2, 4, 4, 9, 1, 5, 4, 2, 1, 9, 2, 9, 4, 0, 9, 6, 8, 0, 3, 2, 9, 4, 4, 1, 0, 8, 0, 1, 2, 4, 7, 1, 3, 8, 6, 9, 4, 8, 5, 4, 3, 2, 2, 5, 1, 2, 9, 6, 6, 5, 4, 1, 3, 2, 3, 3, 2, 7, 9, 2, 6, 9, 5, 3, 9, 1, 2, 7, 4, 5, 5, 1, 6, 0, 4, 9, 1, 0, 4, 7, 7, 8, 9, 1, 8, 7, 2

Offset: 0

Jean-François Alcover, May 27 2014

			0.808652518350212244915421929409680329441...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Renyi's parking constant, p. 279.

Eric Weisstein's MathWorld, Rényi's Parking Constants

Cf. A050996.

digits = 102; NIntegrate[(2*x + 1)*Exp[-2*(x + Exp[-x] - 1)]*Exp[-2*(-ExpIntegralEi[-x] + Log[x] + EulerGamma)], {x, 0, Infinity}, WorkingPrecision -> digits + 5] // RealDigits[#, 10, digits] & // First

Integral_{x>=0} (2*x+1)*exp(-2*(x+exp(-x)-1))*exp(-2*(-Ei(-x)+log(x)+gamma)) dx, where Ei is the exponential integral function and gamma the Euler-Mascheroni constant.

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.

Original entry on oeis.org

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

Jean-François Alcover, Sep 16 2014

			0.0381563991904265053291044982253...

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

Eric Weisstein's MathWorld, Rényi's Parking Constants

Cf. A050996.

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]]]

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.

A247847 Decimal expansion of m = (1-1/e^2)/2, one of Renyi's parking constants.

Original entry on oeis.org

4, 3, 2, 3, 3, 2, 3, 5, 8, 3, 8, 1, 6, 9, 3, 6, 5, 4, 0, 5, 3, 0, 0, 0, 2, 5, 2, 5, 1, 3, 7, 5, 7, 7, 9, 8, 2, 9, 6, 1, 8, 4, 2, 2, 7, 0, 4, 5, 2, 1, 2, 0, 5, 9, 2, 6, 5, 9, 2, 0, 5, 6, 3, 6, 7, 2, 9, 6, 3, 3, 1, 2, 9, 4, 9, 2, 5, 6, 1, 5, 5, 0, 3, 1, 4, 5, 0, 9, 3, 8, 7, 5, 4, 6, 7, 1, 4, 7, 5, 6, 2, 2, 4, 6

Offset: 0

Jean-François Alcover, Sep 25 2014

Curiously, this Renyi parking constant is very close to the prime generated continued fraction A084255 (gap ~ 10^-7).

			0.432332358381693654053000252513757798296184227045212...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Renyi's parking constant, p. 280.

Eric Weisstein's MathWorld, Rényi's Parking Constants
Marek Wolf, Continued fractions constructed from prime numbers, arxiv.org/abs/1003.4015, pp. 4-5.

Cf. A001113, A050996, A073333, A084255, A092555, A242943, A243266, A247392.

RealDigits[(1 - 1/E^2)/2 , 10, 104] // First

Define s(n) = Sum_{k = 0..n} 2^k/k!. Then (1 - 1/e^2)/2 = Sum_{n >= 0} 2^n/( (n+1)!*s(n)*s(n+1) ). Cf. A073333. - Peter Bala, Oct 23 2023

A307154 Decimal expansion of the fraction of occupied places on an infinite lattice cover with 3-length segments.

Original entry on oeis.org

8, 2, 3, 6, 5, 2, 9, 6, 3, 1, 7, 7, 3, 3, 8, 3, 3, 6, 9, 0, 0, 6, 7, 1, 8, 7, 7, 8, 1, 1, 6, 4, 7, 8, 8, 7, 2, 1, 3, 9, 2, 3, 6, 6, 2, 0, 5, 3, 9, 2, 9, 8, 6, 8, 0, 9, 1, 4, 3, 7, 2, 3, 5, 0, 0, 7, 1, 8, 2, 2, 0, 1, 8, 0, 9, 8, 1, 2, 0, 0, 7, 9, 0, 9, 0, 5, 5, 8, 9, 2, 6, 4, 8, 7, 4, 0, 3, 0, 3, 3, 7, 1, 9, 6, 3, 8, 5, 4, 5, 9, 2, 8, 8, 9, 7, 9, 3, 3, 4, 2, 4, 8, 8, 7, 7, 2, 1, 2, 7, 1, 9, 6

Offset: 0

Philipp O. Tsvetkov, Mar 27 2019

Solution of the discrete parking problem when infinite lattice randomly filled with 3-length segments.
Solution of the discrete parking problem when infinite lattice randomly filled with 2-length segments is equal to 1-1/e^2 (see A219863).
Also, the limit of a(n) = (3 + 2*(n-3)*a(n-3) + (n-1)*(n-3)*a(n-1))/(n*(n-2)); a(0) = 0; a(1) = 0; a(2) = 0 as n tends to infinity.
If the length of the segments that randomly cover infinite lattice tends to infinity, then the fraction of occupied places is equal to Rényi's parking constant (see A050996).

			0.8236529631773383369006718778116478872139236620539298680914372350071822...

D. G. Radcliffe, Fat men sitting at a bar
Philipp O. Tsvetkov, Stoichiometry of irreversible ligand binding to a one-dimensional lattice, Scientific Reports, Springer Nature (2020) Vol. 10, Article number: 21308.
Eric Weisstein's World of Mathematics, Dawson's Integral

Cf. A050996, A087654, A099288, A219863, A307131, A307132.

evalf(3*sqrt(Pi)*(erfi(2)-erfi(1))/(2*exp(4)), 120) # Vaclav Kotesovec, Mar 28 2019

N[-((3 DawsonF[1])/E^3) + 3 DawsonF[2], 200] // RealDigits

-imag(3*sqrt(Pi)*(erfc(2*I) - erfc(1*I)) / (2*exp(4))) \\ Michel Marcus, May 10 2019

Equals 3*(Dawson(2) - Dawson(1)/e^3).

Equals 3*sqrt(Pi)*(erfi(2) - erfi(1)) / (2*exp(4)).

A096420 Engel expansion of Renyi's Parking constant.

Original entry on oeis.org

2, 3, 3, 3, 3, 10, 10, 12, 29, 29, 47, 175, 757, 90898, 148227, 168544, 2848069, 17425845, 40219759, 49525807, 82271521, 86730226, 100632852, 137736877

Offset: 1

Gerald McGarvey, Aug 08 2004

nonn

Cf. A050996.

A247848 Decimal expansion of m_2 = (2-1/e)/4, one of Renyi's parking constants, the mean car density in case "monomer with nearest neighbor exclusion" for the 2 x infinity strip.

Original entry on oeis.org

4, 0, 8, 0, 3, 0, 1, 3, 9, 7, 0, 7, 1, 3, 9, 4, 1, 9, 6, 0, 1, 1, 1, 9, 0, 5, 7, 4, 5, 9, 6, 3, 4, 7, 8, 3, 1, 3, 8, 5, 4, 7, 2, 1, 7, 2, 4, 2, 0, 5, 8, 0, 4, 1, 3, 7, 3, 0, 4, 0, 7, 9, 9, 5, 7, 5, 6, 3, 4, 6, 2, 6, 0, 6, 3, 7, 7, 5, 0, 4, 9, 1, 6, 0, 7, 1, 3, 1, 8, 1, 4, 1, 3, 5, 2, 0, 0, 8, 9, 0, 6, 3, 3, 4, 3

Offset: 0

Jean-François Alcover, Sep 25 2014

			0.408030139707139419601119057459634783138547217242058041373...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.3 Renyi's parking constant, p. 281.

Eric Weisstein's MathWorld, Rényi's Parking Constants

Cf. A050996, A092555, A242943, A243266, A247392, A247847.

RealDigits[(2 - 1/E)/4 , 10, 105] // First

A276515 Decimal expansion of decay rate for random parking and halving.

Original entry on oeis.org

6, 1, 6, 4, 4, 5, 9, 8, 1, 6, 5, 2, 1, 7, 9, 8, 9, 1, 7, 0, 6, 2, 5, 1, 2, 7, 7, 3, 7, 4, 7, 1, 2, 4, 8, 8, 4, 7, 7, 5, 2, 9, 0, 6, 1, 7, 3, 0, 4, 0, 5, 2, 9, 5, 9, 1, 8, 8, 8, 3, 1, 2, 6, 5, 6, 3, 0, 1, 6, 3, 3, 8, 1, 5, 3, 8, 9, 0, 4, 6, 7, 8, 3, 4, 3, 8, 8, 4, 5, 4, 4, 6, 5, 2, 0, 3, 0, 1, 5, 5, 2, 2, 1, 7, 6, 6, 0, 8, 1, 2, 5, 3, 0, 0, 9, 4, 5, 7, 1, 4, 7, 3, 7, 7, 9, 5, 2, 6, 7

Offset: 0

Michael Mackey, Sep 06 2016

The asymptotic geometric rate of decay of an interval in which cars of fixed length are randomly parked to jamming point before the car length is halved and the process repeated.

M. Mackey, W.G. Sullivan, Exhaustion of an interval by iterated Rényi parking, J. Math. Anal. Appl., to appear (2016).

Cf. A050996.

A307184 Decimal expansion of the fraction of occupied places on an infinite lattice cover with 4-length segments.

Original entry on oeis.org

8, 0, 3, 8, 9, 3, 4, 7, 9, 9, 1, 5, 3, 7, 6, 9, 7, 2, 6, 6, 6, 2, 9, 7, 4, 1, 9, 5, 0, 3, 2, 1, 3, 4, 2, 0, 5, 4, 6, 8, 7, 9, 1, 6, 4, 8, 5, 7, 7, 0, 8, 3, 5, 9, 2, 3, 9, 7, 2, 9, 9, 3, 2, 8, 0, 7, 0, 9, 4, 5, 6, 0, 9, 5, 0, 7, 6, 0, 3, 6, 1, 5

Offset: 0

Philipp O. Tsvetkov, Mar 28 2019

The solution of the discrete parking problem when infinite lattice randomly filled with L-length segments at L=4.
At L=3 it is equal to 3*(Dawson(2) - Dawson(1)/e^3) (see A307154).
At L=2 it is equal to 1-1/e^2 (see A219863).
The general solution of the discrete parking problem when infinite lattice randomly filled with L-length segments is equal to L*e(-2H(L-1))*Integral_{x=0..1} e^(2*(t + t^2/2 + t^3/3 + ... + t^(L-1)/(L-1))) dx, where H(L) is harmonic number.
Also, the limit of the following recurrence as n tends to infinity: a(n) = (4 + 2(n-4)*a(n-4) + (n-1)*(n-4)*a(n-1))/(n*(n-3)); a(0) = 0; a(1) = 0; a(2) = 0; a(3) = 0.
If L tends to infinity, then the fraction of occupied places is equal to Rényi's parking constant (see A050996).

			0.80389347991537697266629741950321342054687916485770835923972993280709456095...

D. G. Radcliffe, Fat men sitting at a bar
Philipp O. Tsvetkov, Stoichiometry of irreversible ligand binding to a one-dimensional lattice, Scientific Reports, Springer Nature (2020) Vol. 10, Article number: 21308.

Cf. A219863, A050996, A307132, A307131, A307154.

evalf(Integrate(4*exp(2*(t + t^2/2 + t^3/3) - 11/3), t= 0..1), 120); # Vaclav Kotesovec, Mar 28 2019

RealDigits[ N[(4*Integrate[E^(2*(t + t^2/2 + t^3/3)), {t, 0, 1}])/E^(11/3), 200]][[1]]

intnum(t=0, 1, 4*exp(2*(t + t^2/2 + t^3/3) - 11/3)) \\ Michel Marcus, May 10 2019

4*Integral_{x=0..1} e^(2*(t + t^2/2 + t^3/3)) dx / e^(11/3).

cp's OEIS Frontend

A243266 Decimal expansion of a parking constant related to the asymptotic expected number of cars, assuming random car lengths.

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A050995 Reduced denominators of series expansion for integrand in Renyi's parking constant.

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A242943 Decimal expansion of the mean car density associated with Solomon's variation in Renyi's one-dimensional parking problem.

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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.

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A247847 Decimal expansion of m = (1-1/e^2)/2, one of Renyi's parking constants.

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A307154 Decimal expansion of the fraction of occupied places on an infinite lattice cover with 3-length segments.

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A096420 Engel expansion of Renyi's Parking constant.

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A247848 Decimal expansion of m_2 = (2-1/e)/4, one of Renyi's parking constants, the mean car density in case "monomer with nearest neighbor exclusion" for the 2 x infinity strip.

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