Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$.

Is there any formula for $\lambda_n(x)$?

I am particularly interested in the following question.

Let $y_n$ be such that $\lambda_n(y_n)=\frac{n}4$. What is the value of $$\lim_{n\to \infty} (y_n)^n$$ if the limit exists?