Well-definedness of the classical Hofstadter Q-sequence

Determine whether the classical Hofstadter Q-sequence defined by Q(1)=1, Q(2)=1, and Q(n)=Q(n−Q(n−1))+Q(n−Q(n−2)) for n≥3 is well-defined for all integers n≥1; equivalently, prove that the recursive arguments n−Q(n−1) and n−Q(n−2) remain positive for all n≥3.

Background

The paper proves that a perturbed Hofstadter-type sequence with an alternating term is well-defined for all n. In contrast, it explicitly notes that the analogous well-definedness property for the original Hofstadter Q-sequence remains unresolved.

Well-definedness here means that the recursion produces values for every n, i.e., the arguments in the nested calls stay positive. The difficulty of establishing this for the classical Q-sequence is emphasized in the literature, and this work highlights that the finite-state reduction enabling the proof in the perturbed case does not appear to extend to the unperturbed sequence.