A356805 Decimal expansion of the unique positive real root of the equation x^x^(x - 1) = x + 1.

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

Offset

1,2

Comments

This constant arises from a well-known linear approximation for real height of the tetration x^^x (for x belonging to (1, 2)), where x^^x indicates the tetration of the real base x having the same height (see Links - Wikipedia).

A valuable method to extend tetration to real numbers, and solving equations as the above, has been introduced in 2006 by Hooshmand in his paper "Ultra power and ultra exponential functions" (see Links - Hooshmand).

Links

Table of n, a(n) for n=1..87.

Mohammad Hadi Hooshmand, Ultra power and ultra exponential functions, Integral Transforms and Special Functions, Volume 17(8), 2006, pp. 549-558.

Wikipedia, Tetration (see in particular "Linear approximation for real heights").

Example

1.85566023196173...

Mathematica

RealDigits[x /. FindRoot[x^(x^(x - 1)) == x + 1, {x, 2}, WorkingPrecision -> 100]][[1]]

Prog

(PARI) solve(x=1, 2, x^x^(x - 1) - x - 1) \\ Michel Marcus, Aug 29 2022

Crossrefs

Cf. A124930, A356562.

Sequence in context: A257436 A201295 A011107 * A318335 A243379 A214174

Adjacent sequences: A356802 A356803 A356804 * A356806 A356807 A356808

Keyword

cons,nonn

Author

Marco Ripà and Flavio Niccolò Baglioni, Aug 28 2022

Status

approved