Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton $X$. Is there a 2 dimensional 3-colorstate (black,white,gray) mobile automaton $Q$ such that when $Q(t)$ is the $t$ th iteration we have :
$$L(Q(t)) = O(f(t))$$
Where $O$ is big-O notation.
Usefull link : http://mathworld.wolfram.com/MobileAutomaton.html