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Decimal expansion of the fraction of occupied places on an infinite lattice cover with 3-length segments.
1

%I #31 Mar 24 2021 22:59:03

%S 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,

%T 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,

%U 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

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

%C Solution of the discrete parking problem when infinite lattice randomly filled with 3-length segments.

%C Solution of the discrete parking problem when infinite lattice randomly filled with 2-length segments is equal to 1-1/e^2 (see A219863).

%C 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.

%C 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).

%H D. G. Radcliffe, <a href="https://mathblag.files.wordpress.com/2012/12/fatmen.pdf">Fat men sitting at a bar</a>

%H Philipp O. Tsvetkov, <a href="https://doi.org/10.1038/s41598-020-77896-0">Stoichiometry of irreversible ligand binding to a one-dimensional lattice</a>, Scientific Reports, Springer Nature (2020) Vol. 10, Article number: 21308.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DawsonsIntegral.html">Dawson's Integral</a>

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

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

%e 0.8236529631773383369006718778116478872139236620539298680914372350071822...

%p evalf(3*sqrt(Pi)*(erfi(2)-erfi(1))/(2*exp(4)), 120) # _Vaclav Kotesovec_, Mar 28 2019

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

%o (PARI) -imag(3*sqrt(Pi)*(erfc(2*I) - erfc(1*I)) / (2*exp(4))) \\ _Michel Marcus_, May 10 2019

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

%K nonn,cons

%O 0,1

%A _Philipp O. Tsvetkov_, Mar 27 2019