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%I #84 Dec 16 2022 09:02:27
%S 1,6,4,8,7,2,1,2,7,0,7,0,0,1,2,8,1,4,6,8,4,8,6,5,0,7,8,7,8,1,4,1,6,3,
%T 5,7,1,6,5,3,7,7,6,1,0,0,7,1,0,1,4,8,0,1,1,5,7,5,0,7,9,3,1,1,6,4,0,6,
%U 6,1,0,2,1,1,9,4,2,1,5,6,0,8,6,3,2,7,7,6,5,2,0,0,5,6,3,6,6,6,4
%N Decimal expansion of sqrt(e).
%C Also where x^(x^(-2)) is a maximum. - _Robert G. Wilson v_, Oct 22 2014
%C e^(1/2) maximizes the value of x^(c/(x^2)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - _A.H.M. Smeets_, Aug 16 2018
%H Harry J. Smith, <a href="/A019774/b019774.txt">Table of n, a(n) for n = 1..20000</a>
%H Michael Penn, <a href="https://www.youtube.com/watch?v=FqifDb1_gfc">On means of binomial coefficients.</a>, YouTube video, 2020.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F sqrt(e) = Sum_{n>=0} 1/(2^n*n!) = Sum_{n>=0} 1/(2n)!!. - _Daniel Forgues_, Apr 17 2011
%F sqrt(e) = 1 + Sum_{n>0} Product_{i=1..n} 1/(2n). - _Ralf Stephan_, Sep 11 2013
%F Continued fraction representation: sqrt(e) = 1 + 1/(1 + 2/(3 + 4/(5 + ... ))). See A000354 for details. - _Peter Bala_, Jan 30 2015
%F sqrt(e) = (1/2)*( 1 + (3 + (5 + (7 + ...)/6)/4)/2 ) = 1 + (1 + (1 + (1 + ...)/6)/4)/2. - _Rok Cestnik_, Jan 19 2017
%F sqrt(e) = 2*Sum_{n >= 0} 1/((1 - 4*n^2)*(2^n)*n!). - _Peter Bala_, Jan 16 2022
%F sqrt(e) = (16/31)*(1 + Sum_{n>=1} (1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!). - _Alexander R. Povolotsky_, Jul 01 2022
%e 1.6487212707001281468486507878141635716537761007101480115750...
%p evalf(sqrt(exp(1)), 120); # _Muniru A Asiru_, Aug 16 2018
%t RealDigits[N[Sqrt[E],200]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 21 2011 *)
%o (PARI) default(realprecision, 20080); x=sqrt(exp(1)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019774.txt", n, " ", d)); \\ _Harry J. Smith_, May 01 2009
%Y Cf. A001113, A058281 for continued fraction for sqrt(e).
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_
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