We studied numerically the frequency spectrum of photons in a multilayered microsphere coated by a quasiperiodic (Pascal) dielectric stack. It is found that the transmittancy spectrum of such a stack consists of quasiband gaps and narrow resonances caused by re-reflection of optical waves. When the number (Pascal order) of layers increases, the band gaps and resonances split, and the structure of the frequency spectrum acquires a fractal form. Some parts of the spectrum show the self-similarity in the different frequency scales.