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A202346 Decimal expansion of x > 0 satisfying 2*x + 2 = exp(x).

%I A202346 #14 Nov 21 2024 15:38:23
%S A202346 1,6,7,8,3,4,6,9,9,0,0,1,6,6,6,0,6,5,3,4,1,2,8,8,4,5,1,2,0,9,4,5,2,3,
%T A202346 0,8,4,8,2,4,4,5,8,7,6,5,3,5,1,6,0,2,2,1,6,3,9,8,3,4,1,8,6,8,3,9,9,0,
%U A202346 4,7,6,4,5,6,8,5,7,1,3,4,3,6,9,9,7,4,6,9,8,2,4,1,8,8,1,2,0,6,3
%N A202346 Decimal expansion of x > 0 satisfying 2*x + 2 = exp(x).
%C A202346 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202346 G. C. Greubel, <a href="/A202346/b202346.txt">Table of n, a(n) for n = 1..10000</a>
%H A202346 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A202346 x<0:  -0.76803904701346556525568352607754...
%e A202346 x>0:  1.6783469900166606534128845120945230...
%t A202346 u = 2; v = 2;
%t A202346 f[x_] := u*x + v; g[x_] := E^x
%t A202346 Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
%t A202346 r = x /. FindRoot[f[x] == g[x], {x, -.8, -.7}, WorkingPrecision -> 110]
%t A202346 RealDigits[r]  (* A202345 *)
%t A202346 r = x /. FindRoot[f[x] == g[x], {x, 1.6, 1.7}, WorkingPrecision -> 110]
%t A202346 RealDigits[r] (* A202346 *)
%t A202346 RealDigits[-1 - LambertW[-1, -Exp[-1]/2], 10, 100][[1]] (* _G. C. Greubel_, Nov 09 2017 *)
%o A202346 (PARI) solve(x=0, 2, 2*x+2-exp(x)) \\ _Michel Marcus_, Nov 09 2017
%Y A202346 Cf. A202320.
%K A202346 nonn,cons
%O A202346 1,2
%A A202346 _Clark Kimberling_, Dec 17 2011