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

Author

Vladimir Reshetnikov, Apr 30 2013

Keywords

• nonn
• cofr
• easy

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

• 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

Crossrefs

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

Programs

Mathematica

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

Extensions

Offset changed by Andrew Howroyd, Jul 07 2024