Is there a binary sequence that contains all contiguous finite sub-sequences without unnecessary repetition. This was something I've been playing around inside of my head and I'm unsure it has a solution. An infinite sequence S of 1's and 0's that fulfils the following criteria. * Every finite binary sequence is present as a contiguous subsequence within S * A contiguous subsequencence of length N can not be repeated until all other possible contiguous subsequences of that length have been visited.

No. To get all the one bit codes you start with $01$ or its complement. You can extend that to $01100$ and get all the two bit codes. To get the threes you have to go to $011000111$ but you get two $011$s before $111$ and still don’t have $101$