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A202350 Decimal expansion of x > 0 satisfying e*x + 1 = exp(x).

%I A202350 #29 Feb 07 2025 16:44:07
%S A202350 1,7,5,0,7,8,6,7,2,2,6,8,0,1,4,6,3,6,7,5,7,0,0,1,4,8,7,7,2,5,5,3,3,2,
%T A202350 8,9,4,1,3,7,8,6,6,3,4,9,4,0,8,2,6,8,4,9,0,8,0,5,9,4,5,7,5,6,1,6,0,8,
%U A202350 4,7,8,6,1,9,5,5,1,7,3,2,0,6,4,9,0,1,6,9,1,4,7,2,5,8,0,3,7,0,8
%N A202350 Decimal expansion of x > 0 satisfying e*x + 1 = exp(x).
%C A202350 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202350 G. C. Greubel, <a href="/A202350/b202350.txt">Table of n, a(n) for n = 1..10000</a>
%H A202350 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A202350 Equals -LambertW(-1, -1/exp(1+1/e)) - 1/e. - _Andrea Pinos_, Sep 12 2023
%e A202350 1.750786722680146367570014877255332...
%t A202350 u = E; v = 1;
%t A202350 f[x_] := u*x + v; g[x_] := E^x
%t A202350 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A202350 r = x /. FindRoot[f[x] == g[x], {x, 1.7, 1.8}, WorkingPrecision -> 110]
%t A202350 RealDigits[r]  (* A202350 *)
%t A202350 (* alternate program *)
%t A202350 RealDigits[(-1 - E*ProductLog[-1, -E^(-1-1/E)])/E, 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%Y A202350 Cf. A001113, A202320.
%K A202350 nonn,cons
%O A202350 1,2
%A A202350 _Clark Kimberling_, Dec 17 2011