A198683 Number of distinct values taken by i^i^...^i (with n i's and parentheses inserted in all possible ways) where i = sqrt(-1) and ^ denotes the principal value of the exponential function.

1, 1, 2, 3, 7, 15, 34, 77, 187, 462, 1152

Offset: 1

Author

Vladimir Reshetnikov, Oct 28 2011

Keywords

• nonn
• more
• nice

Comments

There are C(n-1) ways of inserting the parentheses (where C is a Catalan number, A000108), but not all arrangements produce different values.

At n=10, the expression i^(i^(((i^i)^i)^(i^((i^i)^(i^i))))) evaluates to a large complex number, C = -6.795047376...*10^34 - i*6.044219499...*10^34; as a result, i^C, which arises at n=11, is very large, having a magnitude of e^((-Pi/2)*(-6.044219499...*10^34)) = 4.1007...*10^41232950809707420597749203381002924. - Jon E. Schoenfield, Nov 21 2015

Note that if a is a REAL positive number, the number of different values of a^a^...^a with n a's is at most A000081(n). But this relies on the identity (x^y)^z = (x^z)^y = x^(yz), which is not always true for complex numbers with the principal value of the power function. Thus if Y = ((i^i)^i)^i, we have (i^i)^Y / (i^Y)^i = exp(-2 Pi). - Robert Israel, Nov 27 2015 [So for the present sequence, we know a(n) <= A000108(n-1), but we do not know that a(n) <= A000081(n). - N. J. A. Sloane, Nov 28 2015]

Examples

			a(1) = 1: there is one value, i.
a(2) = 1: there is one value, i^i = exp(i Pi / 2)^i = exp(-Pi/2) = 0.2079...
a(3) = 2: there are two values: (i^i)^i = i^(-1) = 1/i = -i and i^(i^i) = i^0.2079... = exp(0.2079... i Pi / 2) = 0.9472... + 0.3208... i.
a(4) = 3: there are 5 possible parenthesizations but they give only 3 distinct values: i^(i^(i^i)), i^((i^i)^i) = ((i^i)^i)^i, (i^i)^(i^i) = (i^(i^i))^i.
		

Links

• R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
• R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
• MathOverflow, Discussion of related questions
• Jon E. Schoenfield, Tables for n = 1..11 listing all A000108(n-1) ways of inserting the parentheses and identifying the ways that do not yield duplicated values

Crossrefs

Cf. A000081, A000108, A002845, A049006, A077589, A077590.

Programs

Mathematica

iParens[1] = {I}; iParens[n_] := iParens[n] = Union[Flatten[Table[Outer[Power, iParens[k], iParens[n - k]], {k, n - 1}]], SameTest -> Equal]; Table[Length[iParens[n]], {n, 10}]

Extensions

a(11) and a(12) (unconfirmed) from Alonso del Arte, Nov 17 2011

a(12) is said to be either 2919 or 2926. The value will not be included in the data section until it has been confirmed. - N. J. A. Sloane, Nov 26 2015