Recursive formula

Hello,
i want to know if it is ok to build the circuit in a recursive way, but the formula is not the same as what is given at the PDF file. (the circuit output is the same and done in recuresive way, but the input to the smaller circuit is diffirent)

One goal of the project is to clarify how a recursive definition translates to a digital circuit. Please follow the recursion we provided - it is easy.