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
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
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