A219863 Decimal expansion of 1 - 1/e^2.

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

Offset: 0

Author

Charles R Greathouse IV, Nov 30 2012

Keywords

• nonn
• cons

Comments

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 unjoined 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 0.85.

(A 1,k configuration is a substituent branching off a carbon atom, k-2 intermediate carbon atoms, and substituent branching off a carbon atom.)

Solution of the discrete parking problem when infinite lattice randomly filled with 2-length segments. - Philipp O. Tsvetkov, Mar 27 2019

Examples

			0.8646647167633873081060005...
		

References

• Pavel L. Krapivsky, Sidney Redner, and Eli Ben-Naim, A Kinetic View of Statistical Physics, Cambridge University Press, 2010.

Links

• Paul J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers, Journal of the American Chemical Society 61:6 (1939), pp. 1518-1521.
• Philipp O. Tsvetkov, Stoichiometry of irreversible ligand binding to a one-dimensional lattice, Scientific Reports, Springer Nature (2020) Vol. 10, Article number: 21308.
• Index entries for transcendental numbers

Crossrefs

Cf. A092553, A013301, A247847.

Programs

Mathematica

RealDigits[1 - E^(-2), 10, 105][[1]] (* Alonso del Arte, Dec 04 2012 *)

PARI

1-exp(-2)

Formula

Equals 2*A247847. - Philipp O. Tsvetkov, Mar 27 2019