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 A202348 Decimal expansion of x satisfying x = exp(x-2). 12
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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:     u       v        x   -----    --     -------     1      -2     A202348     1      -3     A202494    -1      -1     A202357    -1      -2     A202496    -2      -2     A202497    -2       0     A202498    -3       0     A202499    -Pi      0     A202500   -Pi/2     0     A202501   -2*Pi    -1     A202495 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. 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. 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 LINKS FORMULA Equals -LambertW(-exp(-2)) = 2 - A202320. - Jianing Song, Dec 30 2018 EXAMPLE x = 0.158594339563039362153395341987513893949... MATHEMATICA (* Program 1: A202348 *) u = 1; v = -2; f[x_] := x; g[x_] := E^(u*x + v) Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}] r = x /. FindRoot[f[x] == g[x], {x, .15, .16}, WorkingPrecision -> 110] RealDigits[r]  (* A202348 *) (* Program 2: implicit surface of x=e^(ux+v) *) f[{x_, u_, v_}] := x - E^(u*x + v); t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, .3}]}, {v, 1, 5}, {u, -5, -.1}]; ListPlot3D[Flatten[t, 1]] (* for A202348 *) RealDigits[-ProductLog[-1/E^2], 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *) PROG (PARI) solve(x=0, 1, exp(x-2)-x) \\ Charles R Greathouse IV, Feb 26 2013 CROSSREFS Cf. A202320, A226572. Sequence in context: A246903 A213022 A198732 * A273817 A073212 A059742 Adjacent sequences:  A202345 A202346 A202347 * A202349 A202350 A202351 KEYWORD nonn,cons AUTHOR Clark Kimberling, Dec 20 2011 EXTENSIONS Digits from a(93) on corrected by Jean-François Alcover, Feb 26 2013 STATUS approved

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Last modified June 19 19:11 EDT 2019. Contains 324222 sequences. (Running on oeis4.)