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A202343 Decimal expansion of x > 0 satisfying 2*x + 1 = exp(x).

%I A202343 #19 Feb 07 2025 16:44:07
%S A202343 1,2,5,6,4,3,1,2,0,8,6,2,6,1,6,9,6,7,6,9,8,2,7,3,7,6,1,6,6,0,9,2,1,6,
%T A202343 3,2,6,9,1,6,4,1,6,8,3,1,7,0,1,3,2,3,7,1,1,1,2,5,8,9,4,7,2,7,0,4,8,3,
%U A202343 0,0,4,7,8,5,4,1,0,5,1,9,0,3,5,3,3,6,6,4,7,5,0,9,4,7,2,5,0,8,4
%N A202343 Decimal expansion of x > 0 satisfying 2*x + 1 = exp(x).
%C A202343 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202343 G. C. Greubel, <a href="/A202343/b202343.txt">Table of n, a(n) for n = 1..10000</a>
%H A202343 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A202343 x = 1.25643120862616967698273761660921...
%t A202343 u = 2; v = 1;
%t A202343 f[x_] := u*x + v; g[x_] := E^x
%t A202343 Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
%t A202343 r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t A202343 RealDigits[r]  (* A202343 *)
%t A202343 RealDigits[-1/2 - ProductLog[-1, -1/(2*Sqrt[E])], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%o A202343 (PARI) solve(x=1, 2, 2*x+1-exp(x)) \\ _Michel Marcus_, Nov 09 2017
%Y A202343 Cf. A202320.
%K A202343 nonn,cons
%O A202343 1,2
%A A202343 _Clark Kimberling_, Dec 17 2011