Cellular Automata and Fractal Evolution

Cellular Automaton Basics - Number of CA Rules

We have already seen with S states per cell and N cells per nbhd, there are SN neighborhood configurations.
A rule specifies exactly which of the S possible states results from each of the SN neighborhood configurations.
Suppose we number the neighborhood configurations 1 through SN.
Then there are S outcomes for configuration 1 and S outcomes for configuration 2, so S2 combinations of outcomes for configurations 1 and 2.
For example, take S = 2, and write the states as 0 and 1. Then the combinations for the first two neighborhood configurations are
00, 10, 01, 11
Continuing, there are S3 combinations of outcomes for configurations 1, 2, and 3.
For S = 2, here are the combinations for the first three neighborhood configurations
000, 100, 010, 110, 001, 101, 011, 111
Because there are SN neighborhood configurations,
there are S(SN) rules.
How large is this number?

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