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

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

0,1

Comments

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).

Links

Table of n, a(n) for n=0..81.

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.

Formula

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

Example

0.80389347991537697266629741950321342054687916485770835923972993280709456095...

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A248679 A175574 A195342 * A350811 A154214 A330448

Adjacent sequences: A307181 A307182 A307183 * A307185 A307186 A307187

Keyword

nonn,cons

Author

Philipp O. Tsvetkov, Mar 28 2019

Status

approved