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A202320 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.

%I A202320 #50 Feb 03 2025 10:47:45
%S A202320 1,8,4,1,4,0,5,6,6,0,4,3,6,9,6,0,6,3,7,8,4,6,6,0,4,6,5,8,0,1,2,4,8,6,
%T A202320 1,0,6,0,5,0,3,7,1,3,1,4,3,7,7,6,3,9,6,6,9,5,6,4,8,5,0,0,8,9,5,4,8,1,
%U A202320 8,4,0,8,1,2,1,8,3,1,7,0,0,0,5,1,0,3,4,5,6,7,1,6,9,1,3,0,4,0,1
%N A202320 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.
%C A202320 For many choices of u and v, there is just one x < 0 and one x > 0 satisfying u*x + v = exp(x). Guide to related sequences, with graphs included in Mathematica programs:
%C A202320 u.... v.... least x.....greatest x
%C A202320 1.... 2.... A202320.... A202321
%C A202320 1.... 3.... A202324.... A202325
%C A202320 2.... 1.... ..(x=0).... A202343
%C A202320 3.... 1.... ..(x=0).... A202344
%C A202320 2.... 2.... A202345.... A202346
%C A202320 1.... e.... A202347.... A104689
%C A202320 e.... 1.... ..(x=0).... A202350
%C A202320 3.... 0.... A202351.... A202352
%C A202320 Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z = g(u,v) an implicit surface of f.
%C A202320 For an example related to A202320, take f(x,u,v) = u*x + v - exp(x) and g(u,v) = a nonzero solution x of f(x,u,v) = 0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
%C A202320 The solution for u*x + v = exp(x) is -LambertW(-1/(u*exp(v/u))) - v/u. - _Andrea Pinos_, Sep 14 2023
%H A202320 G. C. Greubel, <a href="/A202320/b202320.txt">Table of n, a(n) for n = 1..5000</a>
%H A202320 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function#Applications">Lambert W function. Applications</a>
%H A202320 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A202320 Equals -LambertW(-exp(-2)) - 2. - _Vaclav Kotesovec_, Jan 09 2014
%F A202320 Equals 2 - A202348. - _Jianing Song_, Dec 30 2018
%e A202320 x < 0:  -1.841405660436960637846604658012486...
%e A202320 x > 0:   1.1461932206205825852370610285213682...
%t A202320 (* Program 1:  A202320 and A202321 *)
%t A202320 u = 1; v = 2;
%t A202320 f[x_] := u*x + v; g[x_] := E^x
%t A202320 Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t A202320 r = x /. FindRoot[f[x] == g[x], {x, -1.9, -1.8}, WorkingPrecision -> 110]
%t A202320 RealDigits[r]  (* A202320 *)
%t A202320 r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t A202320 RealDigits[r]  (* A202321 *)
%t A202320 (* Program 2: implicit surface of u*x+v=e^x *)
%t A202320 f[{x_, u_, v_}] := u*x + v - E^x;
%t A202320 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 2, 4}, {u, 2, 4}];
%t A202320 ListPlot3D[Flatten[t, 1]] (* for A202320 *)
%t A202320 RealDigits[-ProductLog[-1/E^2] - 2, 10, 99] // First (* _Jean-François Alcover_, Feb 26 2013 *)
%o A202320 (PARI) solve(x=-2, -1, x + 2 - exp(x)) \\ _Michel Marcus_, Dec 30 2018
%o A202320 (PARI) lambertw(-exp(-2))-2 \\ _Charles R Greathouse IV_, Feb 03 2025
%Y A202320 Cf. A202322, A202348.
%K A202320 nonn,cons
%O A202320 1,2
%A A202320 _Clark Kimberling_, Dec 16 2011