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%I #21 Jan 02 2023 12:30:48
%S 4,2,8,0,1,1,0,3,7,9,6,4,7,2,9,9,2,3,9,0,2,0,4,1,6,9,3,9,1,7,5,1,2,6,
%T 5,5,3,3,7,6,7,1,0,7,3,7,8,0,3,9,3,9,2,9,2,8,5,6,7,5,4,5,9,1,3,3,3,3,
%U 9,2,4,7,5,0,2,3,3,2,9,3,1,5,9,1,0,8,1,6,7,6,4,4,2,5,0,3,0,6,7,1,9,6,5,2,4
%N Decimal expansion of x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted.
%C Order 5 is the smallest order such that pairwise intersections on (0,1) of distinct power tower functions with parentheses inserted exist. The corresponding y value is 0.66337467860163682654502... . The two intersecting functions are x-> (x^(x^x))^(x^x) and x-> x^(x^((x^x)^x)).
%H Alois P. Heinz, <a href="/A199814/b199814.txt">Table of n, a(n) for n = 0..1000</a>
%H Vladimir Reshetnikov, <a href="http://list.seqfan.eu/oldermail/seqfan/2011-November/008376.html">Intersections of x^x^...^x</a>, SeqFan Discussion, Nov 2011.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%F x in (0,1) : x^(x^2)-2*x = 0.
%e 0.42801103796472992390204...
%p f:= x-> (x^(x^x))^(x^x): g:= x-> x^(x^((x^x)^x)):
%p nmax:= 140: Digits:= nmax+10:
%p xv:= fsolve(f(x)=g(x), x=0..0.99):
%p s:= convert(xv, string):
%p seq(parse(s[n+2]), n=0..nmax);
%t x /. FindRoot[x^(x^2) - 2*x == 0, {x, 1/2}, WorkingPrecision -> 110] // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Dec 05 2013 *)
%Y Cf. A000081 (number of distinct power tower functions), A000108 (number of parenthesizations), A199879 (continued fraction), A199880 (Engel expansion).
%K nonn,cons
%O 0,1
%A _Alois P. Heinz_, Nov 10 2011
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