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A356805
Decimal expansion of the unique positive real root of the equation x^x^(x - 1) = x + 1.
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
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
Sequence in context: A257436 A201295 A011107 * A318335 A243379 A214174
KEYWORD
cons,nonn
AUTHOR
STATUS
approved