A202537 Decimal expansion of x satisfying e^x-e^(-2x)=1.

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

Offset

0,1

Comments

If u>0 and v>0, there is a unique number x satisfying e^(ux)-e^(-vx)=1. Guide to related sequences, with graphs included in Mathematica programs:

u.... v.... x

1.... 1.... A002390

1.... 2.... A202537

1.... 3.... A202538

2.... 1.... A202539

3.... 1.... A202540

2.... 2.... A202541

3.... 3.... A202542

1/2..1/2... A202543

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 A202537, take f(x,u,v)=e^(ux)-e^(-vx)-1 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

Table of n, a(n) for n=0..98.

Example

x=0.382245085840035641329358499184857393759416422...

Mathematica

(* Program 1:  A202537 *)
u = 1; v = 2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .3, .4}, WorkingPrecision -> 110]
RealDigits[r]  (* A202537 *)
(* Program 2: implicit surface for e^(ux)-e(-vx)=1 *)
f[{x_, u_, v_}] := E^(u*x) - E^(-v*x) - 1;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, 1, 4}, {u, 2, 20}];
ListPlot3D[Flatten[t, 1]] (* for A202537 *)
First[ RealDigits[ Log[ Root[#^3 - #^2 - 1 & , 1]], 10, 99]] (* Jean-François Alcover, Feb 26 2013 *)

Prog

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

Crossrefs

Cf. A002390.

Sequence in context: A280835 A335930 A195426 * A356418 A220516 A010627

Adjacent sequences: A202534 A202535 A202536 * A202538 A202539 A202540

Keyword

nonn,cons

Author

Clark Kimberling, Dec 21 2011

Extensions

Digits from a(90) on corrected by Jean-François Alcover, Feb 26 2013

Status

approved