A363229 Decimal expansion of e^(-2*LambertW(-log(2)/4)).

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

Offset: 1

Author

Thomas Scheuerle, May 22 2023

Keywords

• nonn
• cons

Comments

The least real solution of x^2 = 2^sqrt(x). This equation has two real solutions the other is 256.

Let x be this constant, and c = 2*log(x)/log(2); then c^4 = 2^c.

Let x be this constant, and c = 1/sqrt(x); then c^c = 1/2^(1/4).

Examples

			1.5366769...
		

Crossrefs

Cf. A073084, A104750.

Programs

Mathematica

RealDigits[E^(-2 ProductLog[-Log[2]/4]),10,100][[1]]

PARI

\p 200
exp(-2*lambertw(-log(2)/4))

Python

import math; from sympy import LambertW
print([i for i in str("%.30f" % math.exp(-2*LambertW(-math.log(2)/4)))])
# Javier Rivera Romeu, May 22 2023

Formula

Equals e^(-2*Sum_{k>=1} ((-k)^(-1+k)*(-log(2)/4)^k/k!)).

Equals e^(t*log(2)/2) where t = (2^(1/4))^(2^(1/4))^(2^(1/4))^(2^(1/4))^... is the infinite power tower over 2^(1/4).

Equals 16*LambertW(-log(2)/4)^2 / log(2)^2. - Vaclav Kotesovec, May 22 2023