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A202345 Decimal expansion of x < 0 satisfying 2*x + 2 = exp(x).

%I A202345 #14 Nov 21 2024 15:38:12
%S A202345 7,6,8,0,3,9,0,4,7,0,1,3,4,6,5,5,6,5,2,5,5,6,8,3,5,2,6,0,7,7,5,4,7,9,
%T A202345 9,0,9,0,6,8,4,9,1,4,8,8,7,1,9,1,8,1,9,4,5,1,0,3,1,0,3,2,7,2,4,8,3,7,
%U A202345 8,8,9,0,1,2,7,6,2,3,4,2,0,7,0,9,1,4,5,1,3,9,0,2,0,3,3,9,5,2,6
%N A202345 Decimal expansion of x < 0 satisfying 2*x + 2 = exp(x).
%C A202345 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202345 G. C. Greubel, <a href="/A202345/b202345.txt">Table of n, a(n) for n = 0..10000</a>
%H A202345 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A202345 Equals -1 - lambertw(-exp(-1)/2). - _G. C. Greubel_, Nov 09 2017
%e A202345 x<0:  -0.76803904701346556525568352607754...
%e A202345 x>0:  1.678346990016660653412884512094523...
%t A202345 u = 2; v = 2;
%t A202345 f[x_] := u*x + v; g[x_] := E^x
%t A202345 Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
%t A202345 r = x /. FindRoot[f[x] == g[x], {x, -.8, -.7}, WorkingPrecision -> 110]
%t A202345 RealDigits[r]  (* A202345 *)
%t A202345 r = x /. FindRoot[f[x] == g[x], {x, 1.6, 1.7}, WorkingPrecision -> 110]
%t A202345 RealDigits[r] (* A202346 *)
%t A202345 RealDigits[-1 - LambertW[-Exp[-1]/2], 10, 100][[1]] (* _G. C. Greubel_, Nov 09 2017 *)
%o A202345 (PARI) solve(x=-1, 0, 2*x+2-exp(x)) \\ _Michel Marcus_, Nov 09 2017
%Y A202345 Cf. A202320.
%K A202345 nonn,cons
%O A202345 0,1
%A A202345 _Clark Kimberling_, Dec 17 2011