%I
%S 1,1,2,4,8,17,38,88,206,497,1212
%N Take a string of n x's and insert n1 ^'s and n1 pairs of parentheses in all possible legal ways. Sequence gives number of distinct values when x = sqrt(2).
%C For n=9, largest value is x^(x^(x^(x^(x^6)))) and results from the 132nd tree {0,{0,{0,{{{{{{0,0},0},0},0},0},0}}}} or {1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0}.
%H F. Goebel and R. P. Nederpelt, <a href="http://www.jstor.org/stable/2316312">The number of numerical outcomes of iterated powers</a>, Amer. Math. Monthly, 80 (1971), 10971103.
%H R. K. Guy and J. L. Selfridge, <a href="http://www.jstor.org/stable/2319392">The nesting and roosting habits of the laddered parenthesis</a>, Amer. Math. Monthly 80 (8) (1973), 868876.
%H R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a> (annotated cached copy)
%e For n = 4 there are 5 functions: f1(x) = ((x^x)^x)^x; f2(x) = (x^(x^x))^x; f3(x) = x^((x^x)^x); f4(x) = x^(x^(x^x)); f5(x) = (x^x)^(x^x); but only 4 different values when x = sqrt(2).
%Y Cf. A003019, A000081, A002845, A003018, A082543.
%K nonn,more
%O 1,3
%A _W. Edwin Clark_ and _Wouter Meeussen_, Apr 29 2003
%E Term 1212 added by _Vladimir Reshetnikov_, Oct 29 2011
%E a(1) added by _Franklin T. AdamsWatters_, Nov 03 2011
