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# Cycles for Rule 22

In the process of surveying Rule 22, the following table of cycles N (row number) versus periods P (column number) was found. In counting the number of rings with each period, the zero ring was discounted because it occurs for every value of N; likewise a ring was not repeated if its period was a proper divisor of P, nor was redundancy due to shifting or reflection included (thus it is a table of cycle classes).

```      | 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
---+-----------------------------------------------------------
1 | .  1  .  1  .  1  .  1  .  1  .  1  .  1  .  1  .  1  .  1
2 | .  .  .  1  .  .  .  1  .  .  .  1  .  .  .  1  .  .  .  1
3 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
4 | .  .  .  .  .  .  .  1  1  2  1  1  .  .  .  1  1  3  3  5
5 | .  .  .  .  .  .  .  .  .  .  1  1  1  .  .  .  .  .  .  .
6 | .  .  .  .  .  .  .  1  .  1  .  .  .  .  .  1  .  .  .  2
7 | .  .  .  .  .  .  1  .  .  .  .  .  .  1  .  .  .  .  .  .
8 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1
9 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
10 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
11 | .  .  .  .  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .
12 | .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  2  1  2  1  1
13 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
14 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .  .
15 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
16 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
17 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
18 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .
19 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
20 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .  .  .
```

```      |15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
---+-----------------------------------------------------------
21 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
22 | .  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .
23 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
24 | .  .  .  1  .  1  .  .  .  1  .  .  .  .  .  .  .  .  .  1
25 | .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .  .  .  .
26 | .  .  1  .  .  .  .  .  .  .  1  .  1  .  .  .  .  .  .  1
27 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
28 | .  .  .  .  1  .  .  .  .  1  1  3  1  3  .  .  .  3  3  6
29 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .  .  .
30 | .  .  .  .  .  .  .  .  .  .  .  .  1  1  1  .  .  1  .  .
31 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .
32 | .  .  .  .  .  .  .  .  .  1  .  .  .  .  .  .  .  1  .  .
33 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
34 | .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .
```

In addition, there were many cycles in the range with longer periods than convenient for a table, which are listed below.

Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx