Template:Sequence of the Day for January 5

Intended for: January 5, 2012

Timetable

• First draft entered by Alonso del Arte on November 3, 2011 ✓
• Draft to be reviewed by November 5, 2011
• Draft to be approved by December 5, 2011

Yesterday's SOTD * Tomorrow's SOTD

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A000081: Number of rooted trees with

n

nodes (or connected functions with a fixed point).

{ 1, 1, 2, 4, 9, 20, 48, 115, ... }

My interest in this sequence today has to do not with trees or anything in graph theory, but with the following number-theoretical problem:

Given

n

instances of some constant

c

with exponentiation carets and parentheses inserted in every valid way, how many distinct values are produced by the resulting expressions? The answer depends on what

c

is. With four instances of

c = 1 2

, there are four distinct values, but with four instances of

c = i

(the imaginary unit), only three distinct values are produced. But what if you don’t know what

c

is? Is there an upper bound for

n

instances of

c

? This sequence is the answer.