The distribution of elements in automatic double sequences

Let A = (A (i, j))(i, j = 0)(infinity) be a q-automatic double sequence over a finite set Omega. Let g is an element of Omega 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 N-g (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 N-g (A, n) to a certain Lyapunov exponent. In some cases we determine this exponent exactly. (c) 2005 Elsevier B.V. All rights reserved.