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the first example, the simplest is 0,1,1,0

the rules of operation are that you start at each position and take a pair of numbers. At the end, you wrap to get the last member of the last pair.

In the simple series, you get 01,11,10,00. This is every possible pair of numbers.

They are 'compressed' in a series that only requires n elements but produces all n possible pairs.

The next exampe is 0,0,0,1,1,1,0,1

When taken in triplets, you get 000,001,011,111,110,101,010,100 (wrapping as needed)

This is every possible triple of numbers. They are represented by a string of only 8 numbers. So the 'compressed' string of n numbers yields all n * 3 triples.

There should be a string, then, which is 256 characters long which produces all 256 8 character words, I think.

It's not quite a gray code, because 2 bits can change at each step, but I haven't figured out a formula to generate these strings.

There is something interesting about exponents becoming geometric that fascinates me and I wondered does anybody know anything about this or have thoughts on it?