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Take the sequence 001 and repeatedly append its second half to itself, using the larger half if the length is odd. This gives you 00101 then 00101101 then 001011011101 then 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 0010.

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?

Related Questions:

Do runs of every length occur in this string?

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)

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    The number of characters added each iteration follows http://oeis.org/A073941 or http://oeis.org/A005428. It seems that the parity of these numbers may be playing a part in how counting the groups of adjacent ones does not generate an obvious pattern. The parity of A073941(n) is at http://oeis.org/A082416. – brogrenkp Jul 10 '14 at 04:38
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    The restriction to binary digits seems artificial. I'd think it'd be better to consider the generic case (i.e. a string "$a_1,/,a_2/,...,a_n$") and then only afterwards suppose that some of the $a_n$ are identical. – Semiclassical Jul 19 '14 at 18:08
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    Okay, I have been trying to find more connections between this sequence and other things in mathematics. In this comment I will include a bit of how I have been approaching it lately and maybe it will help discover more connections. Make a list of the characters that are added during each iteration: {01, 101, 1101, 011101, 101011101, ...}. Note that the ends of these sequences are the same, because each iteration you add all of what was added before plus some. So reverse each element and note the index at which zeros appear: {1,5,7,10,15,19,22,26,28,32,36,38,41,46...}. – brogrenkp Jul 20 '14 at 01:31
  • Let A be the sequence in my previous comment. Now, going back to your sequence in the post. Let B be A005428 starting with 2 {2,3,4,6,9,14,...}, the number of characters added at each iteration. The last character at each iteration is at position A061419(n)-1. These will always be 1 - which corresponds to 0 not appearing in A. This is b.c. they can be written as 2 plus consecutive sums from B. For example 26 = 2+(2+3+4+6+9). 1 is in A b.c. (for example) the only way to write 25 in this way is 1+(2+3+4+6+9). It came from the character at index 1 in "001", which is 0. – brogrenkp Jul 20 '14 at 02:19
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    I've posted a "spin-off" question: Do runs of every length occur in this string?. – r.e.s. Jul 25 '14 at 13:04
  • @r.e.s. Thanks. That's anther great related question! – Calvin's Hobbies Jul 25 '14 at 14:08
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    Just an idea, not an answer: I think to have a better relation to usual mathematical expressions and make the problem more "algebraic", one could mirror the initial sequence and then prepend instead of append. You initial word "001" is then "100" and the binary interpretation is $a_0=4$. Then pre pending the leading subsequence "10" means adding $(2^1) \cdot (2 \cdot 4)$ to $a_0$ to get $a_1=20$. Then iterate. This might be easier to handle, and possibly results in some Collatz-like iteration – Gottfried Helms Dec 17 '14 at 21:07
  • Just a thought :See the section on Dragon Curves in : Mathematical Recreations And Essays by Rousse-Ball and Coxeter. Finite binary sequences code for a sequence with 2 types of paper folds, which when unfolded, not back to 180 degrees but to 90 or 60, making interesting recursive patterns. – DanielWainfleet Jan 27 '16 at 08:19
  • I will say that it definitely is somewhat significant - if you call something like the look-and-say sequence important. But both manipulate our right-to-left digit systems. It uses our lack of openness to more odd digit notations by stringing digits on to the end. This means that these sequences only have significance if they are connected to something actually important related to the way we write digits. Other than that, I don't see anything. It may help to transfer each number in the sequence to decimal, because we are trained to look at decimal numbers. It would help with finding a pattern – asher drummond Jun 02 '16 at 12:16

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If I use abc to generate the sequence it comes out like this:

abc                       3
bc                        5
c bc                      8
c c bc                    12
bc c c bc                 18
c bc bc c c bc            27
c c c bc c bc bc c c bc   41

The list of numbers on the right, the sequence length up to that point, is A061419. A Mills-type formula for this sequence has the constant $\frac{2}{3}K(3)$, where $K(3)$ is A083286, and this last constant is related to the Josephus problem.

So your sequence is related to prime number sieves and the prime number theorem, which are closely linked with the Josephus problem. That you could not find the sequence in the OEIS is most likely because mathematics in this area lags behind other fields of number theory.

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