We study the electronic structure of a new type of Fibonacci superlattice based on Si delta-doped GaAs. Assuming that delta-doped layers are equally spaced, quasiperiodicity is introduced by selecting two different donor concentrations and arranging them according to the Fibonacci series along the growth direction. The one-electron potential due to delta-doping is obtained by means of the Thomas-Fermi approach. The resulting energy spectrum is then found by solving the corresponding effective-mass wave equation. We find that a self-similar spectrum can be seen in the band structure. Electronic transport properties of samples are also discussed and related to the degree of spatial localization of electronic envelope functions.