I am considering this graph and semi-infinite walks on it. By $W_n$ denote the set of words of length $n$. I am trying to find out how many words of length $n$ there are, i.e. to find the cardinality of $W_n$.
For example, ABCD is a word of length $4$.
This seems to be rather tricky, at least I did not find an answer for general $n\in\mathbb{N}$.
Let $A(n)$ be the number of words of length $n$ ending in $A$ and similarly for all the other letters. You can write a set of coupled recurrences like $E(n)=K(n-1)+D(n-1)$, one for each letter. You can encode this in a spreadsheet with $n$ in rows, the letters in columns, and the recurrence relations in the intersections. $A(1)=1$ and the same for all the letters. Write the recurrence into row $2$ and copy down.
Alternately you can think of a column vector with $A(n), B(n), \ldots N(n)$ as entries and a matrix you multiply on the left to increase $n$ to $n+1$. If you can diagonalize the matrix you can find a closed form in terms of powers of the diagonal entries.