Refined Estimates of the Number of Repetition-Free Boolean Functions in the Full Binary Basis $\{\&,\vee,\oplus,-\}$

We consider repetition-free Boolean functions in the basis $\{\&,\vee,\oplus,-\}$, and prove a formula expressing the number of such functions of $n$ variables as a product of Fibonacci numbers. These products are estimated; as a result, we obtain asymptotic estimates for the number of repetition-free Boolean functions. These estimates involve Euler numbers of second order and can be reduced by well-known methods to the form of an exponential-power series. These estimates can be used to construct the final asymptotics of the number of repetition-free Boolean functions in the full binary basis.