I am working on understanding Kolmogorov Complexity. I've been struggling with the following exercise for quite some time now and would appreciate any inputs.
Show that there exists a constant $c$, such that for each $n \in \mathbb{N} -\{0\},$${K((01)^2}^n) \leq \lceil \log_2(n+1)\rceil +c = \lceil\log_2(\log_2(\frac{|{(01)^2}^n |}{2}))\rceil +c $
Through experimentation, I've come to the conclusion that $|{(01)^2}^n| = 2\cdot2^n \implies \log_2(|{(01)^2}^n|)=n+1$. Why is it wrong to simply substitute this in the above equation like so:
${K((01)^2}^n) \leq \lceil \log_2(n+1)\rceil +c = \lceil\log_2(\log_2(|{(01)^2}^n |))\rceil +c $
Why is the length of the string divided by 2 in the problem statement?