Let $L_n$ be the number of all paths of length n in a directed graph(below). It is needed to find $lim_{n \to \infty}\chi(L_n)/n$ where $\chi(L_n)$ is Hartley information in $L_n$ set. (If I am not mistaken, $\chi(L_n) = log(|L_n|)$)
I tried to find formula for $L_n$. To manage this aim I represented my graph as the adjacency matrix (A): $\begin{array}{|l | r|} \hline 1 & 1 \\ \hline 1 & 0 \\ \hline \end{array}$ Then I found its Jordan canonical normal form($J_A): $\begin{array}{|l | r|} \hline \frac{1 + \sqrt{5}}{2} & 0\\ \hline 0 & \frac{1 - \sqrt{5}}{2}\\ \hline \end{array}$
and the transition matrix(C): $\begin{array}{|l | r|} \hline 1 & \frac{2}{1+\sqrt{5}} \\ \hline 1 & \frac{2}{1- \sqrt{5}} \\ \hline \end{array}$.
I did it with an aim to make next countings more simple, because $A^n = C^{-1}\cdot J_A^n \cdot C$. (1)
Then I found $A^n$ with help of (1) formula and summed all elements of my matrix. This sum is(if I am not mistaken): $\frac{10 \cdot \sqrt{5} + 22}{\sqrt{5}(1+\sqrt{5})^2} \cdot (\frac{1+\sqrt{5}}{2})^n + \frac{8 - 4 \cdot \sqrt{5}}{(\sqrt{5} + 5) \cdot (1 - \sqrt{5}} \cdot (\frac{1 - \sqrt{5}}{2})^n$.
This sum is a number of all n-length paths($L_n$).
Please tell me if my thinkings are right and (if they are right) how to find $lim_{n \to \infty}\chi(L_n)/n$ (I am not very good at finding limits). Thank you in advance.
P.S. I will be also very pleased if you at least tell me how to find my limit.