# Thread: Special property of corresponding sequences

1. ## Special property of corresponding sequences

For any sequence s consisting of 1's and 2's, let r(s) denote the length of the nth run of same symbols in s .

There is a unique nontrivial sequence s such that $\displaystyle s(1) = 1$ and $\displaystyle r(r(s(n))) = s(n)$ for all $\displaystyle n$ :

$\displaystyle s = (1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, . . .)$

$\displaystyle r(s) = (2, 1, 2, 2, 1, 2, 1, 1, 2 , 2 ,1, 2, 2, 1, 1, 2, 1, . . .)$

Question : Prove or disprove that every segment of r(s) is a segment of s.

For example , the initial segment 1121 of s occurs in r(s) beginning at the 14th term.

P.S.

I am interested in hints (not full solution)

2. ## Re: Special property of corresponding sequences

Originally Posted by princeps
Question : Prove or disprove that every segment of r(s) is a segment of s.

For example , the initial segment 1121 of s occurs in r(s) beginning at the 14th term.
The question seems to be about finding segments of r(s) in s, not the other way around.

The sequence r(s) does not need to have the property that the maximum possible length of the nth run is 2.

3. ## Re: Special property of corresponding sequences

Have you any idea how to approach to this problem ?

4. ## Re: Special property of corresponding sequences

Originally Posted by princeps
Have you any idea how to approach to this problem ?
Yes.
Originally Posted by emakarov
The sequence r(s) does not need to have the property that the maximum possible length of the nth run is 2.