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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

Table of n, a(n) for n=0..98.

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 December 8 12:31 EST 2019. Contains 329864 sequences. (Running on oeis4.)