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%I #48 Aug 06 2024 10:32:20
%S 2,7,8,4,6,4,5,4,2,7,6,1,0,7,3,7,9,5,1,0,9,3,5,8,7,3,9,0,2,2,9,8,0,1,
%T 5,5,4,3,9,4,7,7,4,8,8,6,1,9,7,4,5,7,6,5,4,5,3,1,7,8,1,0,5,5,3,5,0,2,
%U 9,3,7,5,4,5,9,9,4,9,8,9,8,1,9,2,0,4,9,8,4,2,8,1,1,2,9,9,4,2,8
%N Decimal expansion of the number x satisfying e*x = e^(-x).
%C See A202322 for a guide to related sequences. The Mathematica program includes a graph.
%D Heine Halberstam and Hans Egon-Richert, Sieve Methods, Dover Publications (2011). See Theorem 2.1.
%H G. C. Greubel, <a href="/A202357/b202357.txt">Table of n, a(n) for n = 0..5000</a>
%H W. Gautschi, <a href="https://citeseerx.ist.psu.edu/pdf/852d953b27f4c309e050feec8c1b11b9beb15b4c">The incomplete Gamma Function since Tricomi</a>, Atti Conv. Lincei 147 (1999) 203, eq. (2.16).
%H H. J. H. Tuenter, <a href="http://arxiv.org/abs/math/0606238">On the generalized Poisson distribution</a>, arXiv:math/0606238 [math.ST], 2006. Published version on <a href="http://dx.doi.org/10.1111/1467-9574.00147">On the Generalized Poisson Distribution</a>, Statistica Neerlandica, 54(3):374-376, November 2000.
%H S. M. Zemyan, <a href="http://www.jstor.org/stable/30037629">On the zeros of the Nth partial sum of the exponential series</a>, Am. Math. Monthly 112 (2005) 891-909.
%F The constant in A202355 minus 1. - _R. J. Mathar_, Dec 21 2011
%F 1+x+log(x)=0. - _R. J. Mathar_, Nov 02 2012
%F Equals LambertW(exp(-1)). - _Vaclav Kotesovec_, Jan 10 2014
%e x=0.2784645427610737951093587390229801554394774886...
%t u = E; v = 0;
%t f[x_] := u*x + v; g[x_] := E^-x
%t Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, .27, .28}, WorkingPrecision -> 110]
%t RealDigits[r] (* A202357 *)
%t RealDigits[ ProductLog[1/E], 10, 99] // First (* _Jean-François Alcover_, Feb 14 2013 *)
%t RealDigits[LambertW[Exp[-1]],10,120][[1]] (* _Harvey P. Dale_, Dec 24 2019 *)
%o (PARI) lambertw(exp(-1)) \\ _Michel Marcus_, Mar 21 2016
%Y Cf. A202322, A215077, A218300.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Dec 18 2011
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