I have tonight been mulling over a lot of mathematics and I was considering, given a sequence can only be generated from elements 0 and 1, e.g: {1,0,0,1,0,1,1,1} does there exist an upper bound on the longest possible sequence such that you have a sequence:

{a

Where there NEVER exists 3 equal subsequences:

{a

Basically a subsequence of elements can not repeat itself 3 times, one after the other e.g {1,1,1} or {0,1,0,1,0,1} would not be allowed, but {1,1,0,1} would be fine.

If there does exist a finite upper bound I wish to extend this to eventually include more elements, but right now I’m quite perplexed about this situation. Is there some mathematics I can learn to help me with this sort of problem?

{a

_{1}, a_{2}, ..., a_{n}}Where there NEVER exists 3 equal subsequences:

{a

_{i}, ..., a_{i+j}} = {a_{i+j+1}, ..., a_{i+2j+1}} = {a_{i+2j+2}, ..., a_{i+3j+2}}Basically a subsequence of elements can not repeat itself 3 times, one after the other e.g {1,1,1} or {0,1,0,1,0,1} would not be allowed, but {1,1,0,1} would be fine.

If there does exist a finite upper bound I wish to extend this to eventually include more elements, but right now I’m quite perplexed about this situation. Is there some mathematics I can learn to help me with this sort of problem?

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