  We revisit the quantized version of the harmonic oscillator obtained through
a q-dependent family of coherent states. For each q, 0< q < 1, these normalized
states form an overcomplete set that resolves the unity with respect to an
explicit measure. We restrict our study to the case in which 1/q is a quadratic
unit Pisot number: the q-deformed integers form Fibonacci-like sequences of
integers. We then examine the main characteristics of the corresponding quantum
oscillator: localization in the configuration and in the phase spaces, angle
operator, probability distributions and related statistical features, time
evolution and semi-classical phase space trajectories.
