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

0,2

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 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

Links

Table of n, a(n) for n=0..104.

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

Formula

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)

Example

0.1353352832366...

Mathematica

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

Prog

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

Crossrefs

Cf. A019774, A001113, A068985, A219863.

Sequence in context: A251754 A225581 A275391 * A112755 A239278 A284721

Adjacent sequences: A092550 A092551 A092552 * A092554 A092555 A092556

Keyword

cons,nonn

Author

Mohammad K. Azarian, Apr 09 2004

Status

approved