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A202321 Decimal expansion of x > 0 satisfying x + 2 = exp(x).

%I A202321 #21 Nov 21 2024 15:37:50
%S A202321 1,1,4,6,1,9,3,2,2,0,6,2,0,5,8,2,5,8,5,2,3,7,0,6,1,0,2,8,5,2,1,3,6,8,
%T A202321 2,5,2,8,8,8,6,6,2,0,4,6,1,8,2,4,8,8,4,2,6,0,3,4,6,1,9,2,9,1,2,8,6,7,
%U A202321 7,5,1,6,3,9,8,7,5,4,8,8,7,0,7,7,4,3,9,6,0,6,6,1,6,9,0,4,4,6,7
%N A202321 Decimal expansion of x > 0 satisfying x + 2 = exp(x).
%C A202321 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202321 G. C. Greubel, <a href="/A202321/b202321.txt">Table of n, a(n) for n = 1..10000</a>
%H A202321 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A202321 a(n) = -LambertW(-1, -exp(-2)) - 2. - _Vaclav Kotesovec_, Jan 09 2014
%e A202321 x < 0: -1.841405660436960637846604658012486...
%e A202321 x > 0:  1.1461932206205825852370610285213682...
%t A202321 u = 1; v = 2;
%t A202321 f[x_] := u*x + v; g[x_] := E^x
%t A202321 Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t A202321 r = x /. FindRoot[f[x] == g[x], {x, -1.9, -1.8}, WorkingPrecision -> 110]
%t A202321 RealDigits[r]  (* A202320 *)
%t A202321 r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t A202321 RealDigits[r]  (* A202321 *)
%t A202321 RealDigits[-ProductLog[-1, -1/E^2] - 2, 10, 99] // First (* _Jean-François Alcover_, Feb 26 2013 *)
%o A202321 (PARI) solve(x=1, 2, x+2-exp(x)) \\ _Michel Marcus_, Nov 09 2017
%Y A202321 Cf. A202320.
%K A202321 nonn,cons
%O A202321 1,3
%A A202321 _Clark Kimberling_, Dec 16 2011