Number of automatas in single output automaton of n inputs

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In John von Neumann's lecture titled Probabilistic Logics and the Synthesis of Reliable Organisms From Unreliable Components there is a defination 1 of single output automaton on page 3. And it is written that number of such automation is $2^{(2n)}$. I have tried to figure out how he got this number but failed. Can someone show how to reach this number ?

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