A202320 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.

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

Offset: 1

Author

Clark Kimberling, Dec 16 2011

Keywords

• nonn
• cons

Comments

For many choices of u and v, there is just one x < 0 and one x > 0 satisfying u*x + v = exp(x). Guide to related sequences, with graphs included in Mathematica programs:

u.... v.... least x.....greatest x

1.... 2.... A202320.... A202321

1.... 3.... A202324.... A202325

2.... 1.... ..(x=0).... A202343

3.... 1.... ..(x=0).... A202344

2.... 2.... A202345.... A202346

1.... e.... A202347.... A104689

e.... 1.... ..(x=0).... A202350

3.... 0.... A202351.... A202352

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 A202320, take f(x,u,v) = u*x + v - exp(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.

The solution for u*x + v = exp(x) is -LambertW(-1/(u*exp(v/u))) - v/u. - Andrea Pinos, Sep 14 2023

Examples

			x < 0:  -1.841405660436960637846604658012486...
x > 0:   1.1461932206205825852370610285213682...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Wikipedia, Lambert W function. Applications
• Index entries for transcendental numbers.

Crossrefs

Cf. A202322, A202348.

Programs

Mathematica

(* Program 1:  A202320 and A202321 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.9, -1.8}, WorkingPrecision -> 110]
RealDigits[r]  (* A202320 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A202321 *)
(* 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, 2, 4}, {u, 2, 4}];
ListPlot3D[Flatten[t, 1]] (* for A202320 *)
RealDigits[-ProductLog[-1/E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *)

PARI

solve(x=-2, -1, x + 2 - exp(x)) \\ Michel Marcus, Dec 30 2018

PARI

lambertw(-exp(-2))-2 \\ Charles R Greathouse IV, Feb 03 2025

Formula

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

Equals 2 - A202348. - Jianing Song, Dec 30 2018