Let W be a finite word on a two symbol alphabet {0,1}; let us say that W is maximal if it is the last item in the list of all its cyclic permutation (ordered lexicographically).


So, for instance:
{0,1} are the maximal words of length 1;
{00, 10, 11} are the maximal words of length 2;
{000, 100, 110, 111} are the maximal words of length 3;
{0000, 1000, 1010, 1100, 1110, 1111} are the maximal words of length 4;
{00000, 10000, 10100, 11000, 11010, 11100, 11110, 11111} are the maximal words of length 5; ...
und so weiter.


Let 
k(n):= number of maximal words of length n


Is there some formula for k(n)?



Minimal words of length n
For aperiodic (sometimes also called, full period) strings, the term you are looking for is Lyndon words. These are the (unique lexicographically-least) representative of a full-period necklace (as stated in the comments, a necklace is the equivalence class under cyclic rotation). The number $k(n)$ you ask for is exactly the number of necklaces, and again, as stated in the comments, it is given by
$k(n)=\frac{1}{n}\sum\_{d|n}\phi(d)2^{n/d}$. You can check out a proof for this in S.W.Golomb's book "Shift Register Sequences" (in the 1967 edition, start looking at around page 171 and look for the cycles of $PCR\_n$).


