Variation of Fermat number, looking for literature on special properties

Question

Suppose we define a recursive function as follows:

$F_k(0) = 1$

For $i \epsilon \mathbb{N}, i>0$, we define:

$F_k(i) = k^{F_k(i-1)}$

So if $k=2$ we get:

$F_2(3) = 2^{2^2}$

$F_2(4) = 2^{{2^2}^2}$

I want to know what such numbers are called, and if there are generalizations of it. Thanks.

Answer 1

Although a variety of notations have been introduced for this, many authors refer to this "iterated exponentiation" as tetration.

Knuth in particular favors an "up-arrow" notation that lends itself to powerful generalization of the idea.

Graham's number $3\uparrow^n 3$ arose in connection with a problem in Ramsey theory, so there are definitely "combinatorial applications of it".