We introduce a new family of number sequences (f (n)) , governed by the recurrence relation f (n) = af (n − un − 1) + bf (n − un − 2), where u = (un)n∈N is a sequence with values 0, 1. Our study focuses on the properties of the sequence of quotients h(n) = f (n + 1)/f (n) and its set of values V(f ) = {h(n) : n ∈ N} for various u. We give a sufficient condition for finiteness of V(f ) and automaticity of (h(n)) , which holds in particular when u is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software Walnut. On the other hand, we prove that the set V(f ) is infinite for other special binary sequences u, and obtain a trichotomy in its topological type when u is eventually periodic.