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Or more importantly, does it have a name?

The function f(k)=x I have found goes under these rules,

  1. There is a group of n strings containing k characters
  2. Each string is 1 character longer than the last. Ex: 1, 22, 333...
  3. Every string cannot contain the strings before it
  4. Characters cannot be repeated more than 3 times in a row
  5. 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.

look at me
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    It is definitely finite. I think you just have to do a search to find the optimum. As you imagine, the number of possibilities grows very rapidly. – Ross Millikan Jan 15 '24 at 21:01
  • If you throw away the condition 4, then it is a function often used as a preparatory step in explanation of how fast the small-tree() function grows (as in: "See how fast is this thing? well, tree() is a great deal faster"). No, it does not seem to have a universally accepted name. – Ivan Neretin Jan 15 '24 at 21:21
  • without condition 4 f(4) is infinite [1,22,343,3443,34443,344443,3444443...] – look at me Jan 16 '24 at 02:50

1 Answers1

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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".

C7X
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