The q-Calkin–Wilf tree

We define a q-analogue of the Calkin–Wilf tree and the Calkin–Wilf sequence. We show that the nth term f(n;q) of the q-analogue of the Calkin–Wilf sequence is the generating function for the number of hyperbinary expansions of n according to the number of powers that are used exactly twice. We also present formulae for branches within the q-analogue of the Calkin–Wilf tree and predecessors and successors of terms in the q-analogue of the Calkin–Wilf sequence.