A219863 -id:A219863 - cp's OEIS frontend

A092553 Decimal expansion of 1/e^2.

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

Consider a substrate (such as polyvinyl alcohol or in forming the polymer of methyl vinyl ketone) in a "1,3 configuration" in which substituents branching off the substrate can irreversibly join with neighboring substituents unless the neighbor is already joined to its other neighbor. Then this constant is the fraction of joined substituents on an infinite substrate.
This also applies to reversible reactions when the rate of forward reaction is much faster than that of backward reaction; see Flory p. 1518 footnote 5. This had "satisfactory accord" with his experimental data using methyl vinyl ketone polymer for which the experimentally-obtained percentage was 0.15.
(A 1,k configuration is a substituent branching off a carbon atom, k-2 intermediate carbon atoms, and substituent branching off a carbon atom.)  - Charles R Greathouse IV, Nov 30 2012
Also the probability, as n increases without bound, that a permutation of length n is simple: no intervals of length 1 < k < n (an interval of a permutation s is a set of contiguous numbers which in s have consecutive indices). - Charles R Greathouse IV, May 14 2014

			0.1353352832366...

M. H. Albert, M. D. Atkinson and M. Klazar, The enumeration of simple permutations, J. Integer Seq. 6 (2003) 03.4.4. arXiv:math/0304213.
R. Brignall, A survey of simple permutations, Permutation Patterns, ed. S. Linton, N. Ruškuc and V. Vatter, Cambridge Univ. Press, 2010, pp. 41—65; arXiv:0801.0963.
Paul J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers, Journal of the American Chemical Society 61:6 (1939), pp. 1518-1521.
Index entries for transcendental numbers

Cf. A019774, A001113, A068985, A219863.

RealDigits[N[1/E^2,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

exp(-2) \\ Charles R Greathouse IV, Nov 30 2012

From Peter Bala, Oct 27 2019: (Start)
1/e^2 = Sum_{k >= 0} (-2)^k/k!.
This is the case n = 0 of the following series acceleration formulas:
1/e^2 = n!*2^n*Sum_{k >= 0} (-2)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*2^(n-k)*binomial(-x,k) are the (unsigned) row polynomials of A137346. Cf. A094816. (End)

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

Original entry on oeis.org

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.

A307132 Denominator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

Original entry on oeis.org

1, 3, 6, 5, 45, 63, 420, 405, 14175, 17325, 187110, 552825, 14189175, 49116375, 729729000, 723647925, 8881133625, 109185701625, 2062396586250, 10257709336875, 428772250281375, 2348038513445625, 53791427762572500, 160789593855515625, 16025362854266390625

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, A307131 (numerators).

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}] // Denominator

Denominator 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.

A013301 E.g.f.: arctanh(arctanh(x)-log(x+1)).

Original entry on oeis.org

0, 1, 6, 150, 7560, 650160, 84823200, 15617281680, 3855823171200, 1229340262550400, 491641405006752000, 241000527467642342400, 142107676844443620710400, 99236357585615999548800000

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

nonn

			1/2!*x^2 + 6/4!*x^4 + 150/6!*x^6 + 7560/8!*x^8 + ...

Cf. A219863.

nn = 20; Table[(CoefficientList[Series[ArcTanh[ArcTanh[x] - Log[1 + x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 05 2015 *)

a(n) ~ (2*n-1)! / (1-exp(-2))^n. - Vaclav Kotesovec, Feb 05 2015

Definition clarified by Vaclav Kotesovec, Feb 05 2015

Prepended missing a(0)=0 from Vaclav Kotesovec, Feb 05 2015

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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A092553 Decimal expansion of 1/e^2.

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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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A307132 Denominator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

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A013301 E.g.f.: arctanh(arctanh(x)-log(x+1)).

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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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