Restricted kk-ary words and functional equations

We present an algorithm for finding a system of recurrence relations for the number of kk-ary words of length nn that satisfy a certain set of conditions. A rewriting of these relations automatically gives a system of functional equations satisfied by the multivariate generating function that counts kk-ary words by their length and the indices of the corresponding recurrence relations. We propose an approach to describing such equations. In several interesting cases the algorithm recovers and refines results on ττ-avoiding kk-ary words and kk-ary words containing ττ exactly once, where ττ is either a classical, a generalized, or a distanced pattern of length three.