A202539 Decimal expansion of the number x satisfying e^(2x)-e^(-x)=1.

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

Offset: 0

Author

Clark Kimberling, Dec 21 2011

Keywords

• nonn
• cons

Comments

See A202537 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			0.281199574322961846512050764067878299792023...
		

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A202537.

Programs

Mathematica

u = 2; v = 1;
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, .2, .3}, WorkingPrecision -> 110]
RealDigits[r]  (* A202539 *)
RealDigits[ Log[ Root[#^3 - # - 1&, 1]], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

PARI

log(polrootsreal(x^3-x-1)[1]) \\ Charles R Greathouse IV, May 15 2019

Formula

Equals log((v^2+12)/(6*v)) with v = (108+12*sqrt(69))^(1/3). - Alois P. Heinz, Jul 14 2022