A formula for the mean length of the longest common subsequence

The expected value $E$ of the longest common subsequence of letters in two random words is considered as a function of the $\alpha=|A|$ of alphabet and of words lengths $m$ and $n$. It is assumed that each letter independently appears at any position with equal probability. A simple expression for $E(\alpha, m, n)$ and its empirical proof are presented for fixed $ \alpha $ and $ m + n $. High accuracy of the formula in a wide range of values is confirmed by numerical simulations.