Let $f(n)$ be the number of binary Lyndon words of length $n$. This sequence is given by OEIS entry A001037. Is it true that $2f(n) \ge f(n+1)$ for all positive $n$?
I have found a general formula to calculate $f(n)$: $$ f(n) = \frac{1}{n} \sum_{d \mid n} \mu(n/d) 2^d. $$ Symmetry also allows us to rewrite it another way: $$ f(n) = \frac{1}{n} \sum_{d \mid n} \mu(d) 2^{n/d}. $$ Here $\mu(x)$ is the Möbius function, and $n$ is a positive integer (the case $n=0$ can be handled separately). It is possible to substitute these identities into the inequality above and perform some simplifications, but it does not seem to go anywhere. Any help is appreciated.
EDIT: My original formula was incorrect, I forgot the factor of $1/n$ in front of the summation.