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%I #34 Jan 25 2023 06:07:56
%S 7,3,5,7,5,8,8,8,2,3,4,2,8,8,4,6,4,3,1,9,1,0,4,7,5,4,0,3,2,2,9,2,1,7,
%T 3,4,8,9,1,6,2,2,2,6,2,0,6,3,5,3,5,6,6,9,0,1,5,6,7,3,6,0,3,3,9,4,9,2,
%U 2,9,9,1,4,8,9,7,9,9,6
%N Decimal expansion of 2/e.
%C From _Johannes W. Meijer_, Jun 27 2016: (Start)
%C This constant is related to the values of zeta(2*n-1) of the Riemann zeta function and the Euler Mascheroni constant gamma. If we define Z(n) = (1/n) * (sum(zeta(2*n-2*k-1) * Z(k), k=0..n-2) + gamma * Z(n-1)), with Z(0) = 1, then limit(Z(n), n -> infinity) = 2/exp(1).
%C Similar formulas appear in A090998 and A112302.
%C The structure of the n! * Z(n) formulas leads to the multinomial coefficients A036039. (End).
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Integral of log x from x = 1/e to e. - _Charles R Greathouse IV_, Apr 16 2015
%F Equals lim_{k->0} 2*(1 - k)^(1/k). - _Ilya Gutkovskiy_, Jun 27 2016
%F Equals Sum_{i>=0} ((-1)^i)(1-i)/i!. - _Maciej Kaniewski_, Sep 10 2017
%F Equals Sum_{i>=0} ((-1)^i)(i^2+2)/i!. - _Maciej Kaniewski_, Sep 12 2017
%F From _Peter Bala_, Mar 21 2022: (Start)
%F 2/e = Integral_{x = 1..oo} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx;
%F 2/e = - Integral_{x = 0..1} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx, both valid for n >= 2. (End)
%e 0.735758882342... = 2*A068985.
%p evalf(2/exp(1)) ; # _R. J. Mathar_, Jul 14 2013
%t RealDigits[2/E,10,120][[1]] (* _Harvey P. Dale_, Dec 25 2013 *)
%o (PARI) 2*exp(-1) \\ _Charles R Greathouse IV_, Apr 16 2015
%Y Cf. A036039, A068985, A112302, A090998.
%K nonn,cons
%O 0,1
%A _Omar E. Pol_, Nov 15 2007
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