Take the sequence
001 and repeatedly append its second half to itself, using the larger half if the length is odd. This gives you
001011011101011101 and so on. Counting the groups of adjacent ones gives
1 2 3 1 3 2 1 3 4 2 1 3 2 3 4.... You get similar results if you start off with other short sequences like
These sequences do not appear to repeat or follow an obvious pattern, yet they are generated from an extremely simple rule, similar to the Thue-Morse sequence or the look-and-say sequence. However, my sequence does not seem to be on OEIS.
I find it hard to believe that I'm the first person to think that appending half of a sequence to itself might be interesting. So my question is do these sequences already have a place in mathematics, however esoteric? If not, then why not? Am I wrong in thinking that this easily defined yet seemingly disorderly sequence is somewhat curious?
Do runs of every length occur in this string? (At Math Overflow)
Where are the runs in this infinite string? (At Programming Puzzles & Code Golf)