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%I #22 Dec 31 2018 03:10:14
%S 1,5,8,5,9,4,3,3,9,5,6,3,0,3,9,3,6,2,1,5,3,3,9,5,3,4,1,9,8,7,5,1,3,8,
%T 9,3,9,4,9,6,2,8,6,8,5,6,2,2,3,6,0,3,3,0,4,3,5,1,4,9,9,1,0,4,5,1,8,1,
%U 5,9,1,8,7,8,1,6,8,2,9,9,9,4,8,9,6,5,4,3,2,8,3,0,8,6,9,5,9,8,3
%N Decimal expansion of x satisfying x = exp(x-2).
%C For many choices of u and v, there is just one value of x satisfying x = exp(u*x+v). Guide to related sequences, with graphs included in Mathematica programs:
%C u v x
%C ----- -- -------
%C 1 -2 A202348
%C 1 -3 A202494
%C -1 -1 A202357
%C -1 -2 A202496
%C -2 -2 A202497
%C -2 0 A202498
%C -3 0 A202499
%C -Pi 0 A202500
%C -Pi/2 0 A202501
%C -2*Pi -1 A202495
%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 this sequence, take f(x,u,v) = x - exp(u*x+v) 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 Actually there are two solutions to x = exp(x-2). This sequence gives the lesser one, x = -LambertW(-exp(-2)), and A226572 gives the greater one, x = -LambertW(-1,-exp(-2)) = 3.14619322062... - _Jianing Song_, Dec 30 2018
%F Equals -LambertW(-exp(-2)) = 2 - A202320. - _Jianing Song_, Dec 30 2018
%e x = 0.158594339563039362153395341987513893949...
%t (* Program 1: A202348 *)
%t u = 1; v = -2;
%t f[x_] := x; g[x_] := E^(u*x + v)
%t Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, .15, .16}, WorkingPrecision -> 110]
%t RealDigits[r] (* A202348 *)
%t (* Program 2: implicit surface of x=e^(ux+v) *)
%t f[{x_, u_, v_}] := x - E^(u*x + v);
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, .3}]}, {v, 1, 5}, {u, -5, -.1}];
%t ListPlot3D[Flatten[t, 1]] (* for A202348 *)
%t RealDigits[-ProductLog[-1/E^2], 10, 99] // First (* _Jean-François Alcover_, Feb 26 2013 *)
%o (PARI) solve(x=0,1,exp(x-2)-x) \\ _Charles R Greathouse IV_, Feb 26 2013
%Y Cf. A202320, A226572.
%K nonn,cons
%O 0,2
%A _Clark Kimberling_, Dec 20 2011
%E Digits from a(93) on corrected by _Jean-François Alcover_, Feb 26 2013
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