A225153 Continued fraction for the positive root of x^x^x^x = 2 (A225134).

1, 2, 4, 5, 1, 1, 184, 1, 1, 8, 1, 7, 1, 12, 3, 1, 4, 2, 1, 2, 1, 125, 1, 2, 1, 1, 2, 2, 5, 12, 7, 1, 8, 2, 1, 6, 1, 3, 2, 1, 2, 1, 14, 1, 1, 1, 3, 1, 1, 6485, 1, 1, 1, 3, 1, 2, 1, 1, 1, 17, 1, 2, 3, 3, 3, 2, 7, 1, 2, 1, 8, 1, 9, 1, 1, 7, 1, 4, 9, 1, 1, 1, 1, 3, 2

Offset

0,2

Comments

x = 1.44660143242986417... = 1 + 1/(2 + 1/(4 + 1/(5 + 1/(1 + 1/(1 + 1/(184 + 1/(...))))))).

This constant is sometimes called the 4th super-root of 2.

It is unknown if it is rational, algebraic irrational, or transcendental. Hence, it is unknown if this continued fraction is aperiodic, or even if it is infinite.

Links

Table of n, a(n) for n=0..84.

J. Marshall Ash and Yiren Tan, A rational number of the form a^a with a irrational, Mathematical Gazette 96, March 2012, pp. 106-109.

Wikipedia, Tetration, Open questions

Wikipedia, Super-root

Mathematica

ContinuedFraction[FindRoot[x^x^x^x == 2, {x, 1}, WorkingPrecision -> 110][[1, 2]], 105]

Crossrefs

Cf. A225134 (decimal expansion), A225208 (Engel expansion), A153510 (second super-root of 2).

Sequence in context: A021412 A258066 A036501 * A360108 A308319 A167380

Adjacent sequences: A225150 A225151 A225152 * A225154 A225155 A225156

Keyword

nonn,cofr,easy

Author

Vladimir Reshetnikov, Apr 30 2013

Extensions

Offset changed by Andrew Howroyd, Jul 07 2024

Status

approved