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%I #37 Oct 02 2022 23:07:01
%S 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,
%T 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,
%U 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
%N Decimal expansion of 1/e^2.
%C 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.
%C 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.
%C (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
%C 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
%H M. H. Albert, M. D. Atkinson and M. Klazar, <a href="http://arxiv.org/abs/math/0304213">The enumeration of simple permutations</a>, J. Integer Seq. 6 (2003) 03.4.4. arXiv:math/0304213.
%H R. Brignall, <a href="http://arxiv.org/abs/0801.0963">A survey of simple permutations</a>, Permutation Patterns, ed. S. Linton, N. Ruškuc and V. Vatter, Cambridge Univ. Press, 2010, pp. 41—65; arXiv:0801.0963.
%H Paul J. Flory, <a href="http://dx.doi.org/10.1021/ja01875a053">Intramolecular reaction between neighboring substituents of vinyl polymers</a>, Journal of the American Chemical Society 61:6 (1939), pp. 1518-1521.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F From _Peter Bala_, Oct 27 2019: (Start)
%F 1/e^2 = Sum_{k >= 0} (-2)^k/k!.
%F This is the case n = 0 of the following series acceleration formulas:
%F 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)
%e 0.1353352832366...
%t RealDigits[N[1/E^2,200]] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2010 *)
%o (PARI) exp(-2) \\ _Charles R Greathouse IV_, Nov 30 2012
%Y Cf. A019774, A001113, A068985, A219863.
%K cons,nonn
%O 0,2
%A _Mohammad K. Azarian_, Apr 09 2004
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