A202322 Decimal expansion of x satisfying x+2=exp(-x).

4, 4, 2, 8, 5, 4, 4, 0, 1, 0, 0, 2, 3, 8, 8, 5, 8, 3, 1, 4, 1, 3, 2, 7, 9, 9, 9, 9, 9, 9, 3, 3, 6, 8, 1, 9, 7, 1, 6, 2, 6, 2, 1, 2, 9, 3, 7, 3, 4, 7, 9, 6, 8, 4, 7, 1, 7, 7, 3, 3, 0, 7, 6, 9, 8, 2, 0, 1, 5, 9, 9, 2, 1, 4, 2, 0, 0, 4, 0, 7, 8, 4, 9, 0, 8, 6, 5, 9, 2, 4, 8, 1, 7, 8, 7, 3, 9, 5, 5

Offset

0,1

Comments

For many choices of u and v, there is just one value of x satisfying u*x+v=e^(-x). Guide to related sequences, with graphs included in Mathematica programs:

u.... v.... x

1.... 2.... A202322

1.... 3.... A202323

2.... 2.... A202353

2.... e.... A202354

1... -1.... A202355

1.... 0.... A030178

2.... 0.... A202356

e.... 0.... A202357

3.... 0.... A202392

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 A202322, take f(x,u,v)=x+2-e^(-x) 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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Index entries for transcendental numbers

Formula

x(u,v) = W(e^(v/u)/u) - v/u, where W = ProductLog = LambertW. - Jean-François Alcover, Feb 14 2013

Equals A226571 - 2 = LambertW(exp(2))-2. - Vaclav Kotesovec, Jan 09 2014

Example

x=-0.442854401002388583141327999999336819716262...

Mathematica

(* Program 1:  A202322 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.45, -.44}, WorkingPrecision -> 110]
RealDigits[r]  (* A202322 *)
(* Program 2: implicit surface of u*x+v=e^(-x) *)
f[{x_, u_, v_}] := u*x + v - E^-x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 1, 3}, {u, 1, 3}];
ListPlot3D[Flatten[t, 1]] (* for A202322 *)
RealDigits[ ProductLog[E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

Prog

(PARI) lambertw(exp(2)) - 2 \\ G. C. Greubel, Jun 10 2017

Crossrefs

Cf. A202320.

Sequence in context: A007525 A151966 A010778 * A365797 A232523 A376878

Adjacent sequences: A202319 A202320 A202321 * A202323 A202324 A202325

Keyword

nonn,cons

Author

Clark Kimberling, Dec 18 2011

Extensions

Digits from a(84) on corrected by Jean-François Alcover, Feb 14 2013

Status

approved