This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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