Property of Fibonacci numbers and the periodiclike perfectly transparent electronic states in Fibonacci chains

We study the properties of Fibonacci numbers and the transparency of clusters for electrons at some values of the energy. For the $m\mathrm{th}$ Fibonacci number $F_m,$ a set of divisors are obtained by $F_m/k=\lfloor F_m/k\rfloor ,$ $1<k<~F_m.$ Interestingly, the numerical and analytical results show that any new divisors of the $m\mathrm{th}$ Fibonacci sequence will appear periodically in the following Fibonacci sequence. Furthermore, in the mixing Fibonacci system, we perform computer simulations and analytical calculations to study the transparent properties and spatial distributions of electronic states with the energies determined by the divisors of Fibonacci systems. The results show that the transmission coefficients are unity and the corresponding wave functions have periodiclike features. We report that an infinite number of one-dimensional disordered lattices, which are composed of some specific Fibonacci clusters, exhibit an absence of localization.