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 A082499 Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible legal ways. Sequence gives number of distinct values when x = sqrt(2). 10

%I

%S 1,1,2,4,8,17,38,88,206,497,1212

%N Take a string of n x's and insert n-1 ^'s and n-1 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), 1097-1103.

%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), 868-876.

%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