# Kolmogorov Complexity, struggle with equation

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?

The statement is equivalent to \begin{align}n+1=\log_2\left(\frac{|(01)^{2^n}|}2\right)&\impliedby |(01)^{2^n}|=2^{n+2}.\end{align} If $$(01)^{2^n}$$ means $$\underbrace{(01)(01)(01)\cdots(01)}_{2^n\,\text{times}}$$ then the statement is incorrect as $$|(01)^{2^n}|=2^{n+1}$$ as you have correctly obtained.