A202348 Decimal expansion of x satisfying x = exp(x-2).

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

Offset: 0

Author

Clark Kimberling, Dec 20 2011

Keywords

• nonn
• cons

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

Examples

			x = 0.158594339563039362153395341987513893949...
		

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A202320, A226572.

Programs

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 *)

PARI

solve(x=0,1,exp(x-2)-x) \\ Charles R Greathouse IV, Feb 26 2013

Formula

Equals -LambertW(-exp(-2)) = 2 - A202320. - Jianing Song, Dec 30 2018

Extensions

Digits from a(93) on corrected by Jean-François Alcover, Feb 26 2013