A356805 - OEIS

%I #13 Sep 05 2022 09:10:16

%S 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,

%T 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,

%U 5,9,6,7,9,7,3,8,7,3,8,5,8,3,4,7,4,8,9

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

%C 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).

%C 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).

%H Mohammad Hadi Hooshmand, <a href="https://doi.org/10.1080/10652460500422247">Ultra power and ultra exponential functions</a>, Integral Transforms and Special Functions, Volume 17(8), 2006, pp. 549-558.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration#Real_heights">Tetration</a> (see in particular "Linear approximation for real heights").

%e 1.85566023196173...

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

%o (PARI) solve(x=1, 2, x^x^(x - 1) - x - 1) \\ _Michel Marcus_, Aug 29 2022

%Y Cf. A124930, A356562.

%K cons,nonn

%O 1,2

%A _Marco Ripà_ and _Flavio Niccolò Baglioni_, Aug 28 2022