A family of q-coherent states is constructed allowing us to obtain a new quantized version of the harmonic oscillator. These q-states are normalized and form an overcomplete set resolving the unity with respect to the appropriate Jackson measure if 0 < q < 1. We only consider here those values of q such that q −1 is a Pisot number. In this case the qdeformed integers ([n]q) form Fibonacci-like sequences of integers. We study the main physical characteristics of the corresponding quantum

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... or: localization in the configuration and in the phase spaces, probability distributions and related statistical features and semi-classical phase space trajectories whose periodicity is related with the fact that q is an algebraic number.