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Consider the list of length $m$ $(1,0,\dots 0)$ we call this list $l_1$, we now define a sequence of lists recursively, where $l_1$ is the previous list, and if $l_n$ is the list $(a_1,a_2\dots a_n)$ then $l_{n+1}$ is the list $(a_1+a_2,a_2+a_3, \dots ,a_{n-1}+a_n,a_n+a_1)$ where all of the operations are done in $\mathbb Z_2$. In other words all the numbers in the lists are 1 or 0 and we use the rules $1+0=0+1=1$ and $1+1=0+0=0$.

The question is whether we can calculate what $l_n$ is for arbitrary values of $m$ and $n$.

Thank you very much in advance.

Regards.

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1 Answer 1

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It seems Rule 102 of the Elementary cellular automaton with wrapped borders:

central cell
 |  next generation
 v  v
000 0
001 1
010 1
011 0
100 0
101 1
110 1
111 0

01100110 = 102

On an unbounded array, it is precisely the SierpiƄski sieve; for more details see Rule 102 from Mathworld.

enter image description here
The first 80 generations for $m = 39$

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  • $\begingroup$ thanks, is there a simple way to know what generation $n$ with an arbitrary $m$ is? $\endgroup$
    – Gorka
    Aug 30, 2014 at 0:53

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