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%I #48 Oct 09 2023 03:39:01
%S 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 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 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 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.
%C 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 u.... v.... least x.....greatest x
%C 1.... 2.... A202320.... A202321
%C 1.... 3.... A202324.... A202325
%C 2.... 1.... ..(x=0).... A202343
%C 3.... 1.... ..(x=0).... A202344
%C 2.... 2.... A202345.... A202346
%C 1.... e.... A202347.... A104689
%C e.... 1.... ..(x=0).... A202350
%C 3.... 0.... A202351.... A202352
%C 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 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 The solution for u*x + v = exp(x) is -LambertW(-1/(u*exp(v/u))) - v/u. - _Andrea Pinos_, Sep 14 2023
%H G. C. Greubel, <a href="/A202320/b202320.txt">Table of n, a(n) for n = 1..5000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function#Applications">Lambert W function. Applications</a>
%F Equals -LambertW(-exp(-2)) - 2. - _Vaclav Kotesovec_, Jan 09 2014
%F Equals 2 - A202348. - _Jianing Song_, Dec 30 2018
%e x < 0: -1.841405660436960637846604658012486...
%e x > 0: 1.1461932206205825852370610285213682...
%t (* Program 1: A202320 and A202321 *)
%t u = 1; v = 2;
%t f[x_] := u*x + v; g[x_] := E^x
%t Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, -1.9, -1.8}, WorkingPrecision -> 110]
%t RealDigits[r] (* A202320 *)
%t r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t RealDigits[r] (* A202321 *)
%t (* Program 2: implicit surface of u*x+v=e^x *)
%t f[{x_, u_, v_}] := u*x + v - E^x;
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 2, 4}, {u, 2, 4}];
%t ListPlot3D[Flatten[t, 1]] (* for A202320 *)
%t RealDigits[-ProductLog[-1/E^2] - 2, 10, 99] // First (* _Jean-François Alcover_, Feb 26 2013 *)
%o (PARI) solve(x=-2, -1, x + 2 - exp(x)) \\ _Michel Marcus_, Dec 30 2018
%Y Cf. A202322, A202348.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Dec 16 2011
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