The distribution of elements in automatic double sequences

Let A=(A(i,j))i,j=0∞ be a q-automatic double sequence over a finite set Ω. Let g∈Ω and assume that the number Ng(A,n) of g's in the nth row of A is finite for each n. We provide a formula for Ng(A,n) as a product of matrices according to the digits in the base q expansion of n. This formula generalizes several results on Pascal's triangle modulo a prime and on recurrence double sequences. It allows us to relate the asymptotic typical behavior of Ng(A,n) to a certain Lyapunov exponent. In some cases we determine this exponent exactly.