login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Take a string of n x's and insert n-1 ^'s and n-2 pairs of parentheses in all possible legal ways. Sequence gives number of distinct values when x = sqrt(2).
13

%I #42 Jan 05 2025 10:11:57

%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-2 pairs of parentheses in all possible legal ways. Sequence gives number of distinct values when x = sqrt(2).

%C For n=10, largest value is 2^(2^127) = x^(x^(x^(x^(x^6)))) = x^(x^(x^((((((x^x)^x)^x)^x)^x)^x))) 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)

%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>

%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).

%t trees[1] = {x};

%t trees[n_] := trees[n] = Flatten@Table[ch1^ch2, {k, n-1}, {ch1, trees[k]}, {ch2, trees[n-k]}];

%t logs[t_] := ((log/@t) //. {log[a_^b_]:>log[a]b, log[a_ b_]:>log[a]+log[b], log[x]->one, log[one]->0});

%t Table[Length@Union[logs@logs@trees[n] /. {one->1, x->Sqrt[2]}, SameTest->Equal], {n, 9}] (* _Andrey Zabolotskiy_, Jan 03 2025 *)

%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 a(11) = 1212 added by _Vladimir Reshetnikov_, Oct 29 2011

%E a(1) added by _Franklin T. Adams-Watters_, Nov 03 2011