A307132 -id:A307132 - cp's OEIS frontend

A307131 Numerator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

1, 2, 5, 4, 37, 52, 349, 338, 11873, 14554, 157567, 466498, 11994551, 41582906, 618626159, 614191052, 7545655031, 92853583996, 1755370057489, 8737266957604, 365468962351379, 2002633668589496, 45904893141293831

Offset: 1

Philipp O. Tsvetkov, Mar 26 2019

The limit of expected fraction of occupied places on n-length lattice randomly filled with 2-length segments at n tends to infinity is equal to 1-1/e^2 (see A219863).

			0, 1, 2/3, 5/6, 4/5, 37/45, 52/63, 349/420, 338/405, 11873/14175, ...

D. G. Radcliffe, Fat men sitting at a bar

Cf. A219863, A231580, A307132 (denominators).

RecurrenceTable[{f[n] == (2 + 2 (n - 2) f[n - 2] + (n - 1) (n - 2) f[n - 1])/(n (n - 1)),f[0] == 0, f[1] == 0}, f, {n, 2, 100}] // Numerator

Numerator of f(n), where f(0)=0; f(1)=0 and f(n) = (2 + 2(n-2)f(n-2) + (n-1)(n-2)f(n-1))/(n(n-1)) for n>1.

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)).

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).

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A307131 Numerator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

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

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