

A297074


Number of ways of inserting parentheses in x^x^...^x (with n x's) whose result is an integer where x = sqrt(2).


0




OFFSET

1,5


COMMENTS

The largest value that can be obtained by inserting parentheses in x^x^x^x^x^x^x^x^x (9 x's), where x = sqrt(2), is x^(x^((((((x^x)^x)^x)^x)^x)^x)) = 2^128 = 340282366920938463463374607431768211456; this is one of the a(9) = 55 ways of inserting parentheses in x^x^x^x^x^x^x^x^x that yield an integer value.


LINKS

Table of n, a(n) for n=1..9.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy, with permission)
Index entries for sequences related to parenthesizing


EXAMPLE

With x = sqrt(2),
x = 1.414213... is not an integer, so a(1) = 0;
x^x = 1.632526... is not an integer, so a(2) = 0.
(x^x)^x = 2 is an integer, but x^(x^x) = 1.760839... is not, so a(3) = 1;
((x^x)^x)^x, (x^x)^(x^x), (x^(x^x))^x, and x^(x^(x^x)) are noninteger values, but x^((x^x)^x) = 2, so a(4) = 1;
the only ways of inserting parentheses in x^x^x^x^x that yield integer values are x^(x^((x^x)^x)) = 2 and (((x^x)^x)^x)^x = 4, so a(5) = 2.


MATHEMATICA

With[{x = Sqrt@ 2}, Array[Count[#, _?IntegerQ] &@ Map[ToExpression@ StringReplace[ToString@ #, {"{" > "(", "}" > ")", ", " > "^"}] &, Groupings[#, 2] /. _Integer > x] &, 9]] (* Michael De Vlieger, Dec 24 2017 *)


CROSSREFS

Cf. A002845, A055113, A082499, A198683.
Sequence in context: A099516 A293741 A291559 * A099963 A152784 A112855
Adjacent sequences: A297071 A297072 A297073 * A297075 A297076 A297077


KEYWORD

nonn,more


AUTHOR

Jon E. Schoenfield, Dec 24 2017


STATUS

approved



