Or more importantly, does it have a name?
The function f(k)=x I have found goes under these rules,
- There is a group of n strings containing k characters
- Each string is 1 character longer than the last. Ex: 1, 22, 333...
- Every string cannot contain the strings before it
- Characters cannot be repeated more than 3 times in a row
- x is the highest possible n that can have a group following these rules
From this I can find
f(1)=1
[1]
f(2)=2
[1, 22]
f(3)=4
[1, 32, 333, 2223]
What do f(4), f(5), f(6) equal?
I don't know how I would calculate this.
When k=5 there is the infinite sequence 1, 22, 333, 2323, 25323, 232423, 2532423, 23242423, 253242423, 2324242423, 25324242423, 232424242423, ... created by setting the first three terms to 1, 22, 333, then expanding $23\underbrace{242424\ldots 2424}_{\textrm{ many copies of }24}23$, inserting a 5 in the middle when necessary. Using blocks "23" and "24" of two characters instead of one keeps this sequence from violating rule 4.
This function looks similar to Friedman's block subsequence function, but different in a few ways, for example Friedman allows "subsequence" to be non-contiguous: e.g. 343 is a subsequence of 3443 in this way, since while you are searching for the first sequence in the second sequence, you are allowed to "skip over" terms in the second sequence. The precise definition is given on page 2 of Friedman's paper "Long Finite Sequences".