  We study the properties of sequences of the energy eigenvalues for some
generalizations of q-deformed oscillators including the p,q-oscillator, the 3-,
4- and 5-parameter deformed oscillators given in the literature. It is shown
that most of the considered models belong to the class of so-called Fibonacci
oscillators for which any three consequtive energy levels satisfy the relation
E_{n+1}=\lambda E_n+\rho E_{n-1} with real constants \lambda, \rho. On the
other hand, for certain \mu-oscillator known from 1993 we prove the fact of its
non-Fibonacci nature. Possible generalizations of the three-term Fibonacci
relation are discussed among which we choose, as most adequate for the
\mu$-oscillator, the so-called quasi-Fibonacci (or local Fibonacci) property of
the energy levels. The property is encoded in the three-term quasi-Fibonacci
(QF) relation with non-constant, n-dependent coefficients \lambda and \rho.
Various aspects of the QF relation are elaborated for the \mu-oscillator and
some of its extensions.
