I liked the sentence “R. K. Guy's research supported by dwindling grant A-4011 of the National Research Council of Canada.” Jim On Sun, Apr 26, 2020 at 8:59 PM Neil Sloane <njasloane@gmail.com> wrote:
Of course there is this famous paper:
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)
which is mentioned in (and visible from) A003018. Incidentally this sequence could use more terms. There is also a link to Math Overflow for "related questions", and a link to the OEIS Index for other sequences involving parenthesizing.
Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA <https://www.google.com/maps/search/11+South+Adelaide+Avenue,+Highland+Park,+NJ+08904,+USA?entry=gmail&source=g> . Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sun, Apr 26, 2020 at 6:39 PM James Propp <jamespropp@gmail.com> wrote:
It's well known that the expression i^i takes on an infinite set of values if we understand w^z to mean any number of the form exp (z (ln w + 2 pi i n)) where ln is a branch of the natural logarithm function.
Since all values of i^i are real, all values of i^i^i (by which I mean i^(i^i))) are on the unit circle, and in fact they form a countable dense subset of the unit circle.
I can't figure out what's going on with i^i^i^i, though. I've posted a Mathematica notebook at http://jamespropp.org/iiii.pdf containing some intriguing images starting on page 2. Each image shows the points in the set of values of i^i^i^i lying in an annulus whose inner and outer radii differ by a factor of 10.
Can anyone see what's going on?
Also, what happens for taller towers of exponentials? Does the set of values of i^i^...^i become dense once the tower is tall enough?
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