An automatic sequence is a letter-to-letter coding of a fixed point of a uniform morphism. More generally, morphic sequences are letter-to-letter codings of fixed points of arbitrary morphisms. There are many examples where an, a priori, morphic sequence with a non-uniform morphism happens to be an automatic sequence. An example is the Lysënok morphism a → aca, b → d, c → b, d → c, the fixed point of which is also a 2-automatic sequence. Such an identification is useful for describing the dynamical systems generated by the fixed point. We give several ways to uncover such hidden automatic sequences, and present many examples. We focus in particular on morphisms associated with Grigorchuk groups.

A generalized Beatty sequence is a sequence V defined by (Formula Presented) where α is a real number, and p, q, r are integers. Such sequences occur, for instance, in homomorphic embeddings of Sturmian languages in the integers. We consider the question of characterizing pairs of integer triples (p, q, r), (s, t, u) such that the two sequences (Formula Presented) are complementary (their image sets are disjoint and cover the positive integers). Most of our results are for the case that α is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals. We also study triples of sequences (Formula Presented) that are complementary in the same sense.