gmc_nxtman wrote:It's pretty interesting that the sequence occurs in an almost-similar form, but not quite. I didn't know that there was already discussion about this particular sequence, I should've probably searched harder.

It might have been hard to find those sequences with the numbers you were looking at:

6, 4, 14, 14, 12, 62, 4, 126, 28, 30, 30, 28, 1022, 24, 126, 124, 4094

So far I haven't found any cases where the sequence generated by one method is really any different from the sequence generated by any of the other methods. At least, my theory is that the Wolfram Cloud list has an error in it in one place (see below) and the other mismatches are just because the nice simple "multiplicative suborder" formula, A160657, fails for a few cases that have subperiods. See below for more detail on that.

Here's the collection of mechanisms that produce these mysterious period numbers:

-- single ON cell in B1/S on a bounded grid

-- 2xN rectangles in the 2x2 rule (B36/S125)

-- Rule 90 repetition periods (see table and question at end of this post)

-- diagonal lines in B2-a3-i/S01c and B2e/S (and no doubt other rule variants).

-- orthogonal dotted lines in B2cek3i/S12cei or variants

*(looks like B2c/S is all that's needed to get the oscillators -- maybe AbhpzTa used B2cek3i/S12cei because it also supports the gliders and spaceships in the "horiship guns" in Golly's Patterns/Non-Totalistic folder)*
Code: Select all

```
x = 55, y = 1, rule = B2c/S
obobobobobobobobobobobobobobobobobobobobobobobobobobobo!
```

-- orthogonal solid lines, rule provided by Blinkerspawn on the ConwayLife Lounge:

B3i4it5ry6k7e/S1e2k3ey4ti5i looks like it duplicates all the periods from the Wolfram Rule 90 list, which is not true for any of the other 2D rules I've collected so far.

-- orthogonal solid lines, making a pattern of 1x2 blocks instead of 2x2 -- hint provided by drc. Here's the mysterious p174762 oscillator again in this rule:

Code: Select all

```
x = 72, y = 1, rule = B2ci3ai4ci8/S02ae3eijkq4iz5a6i7e
72o!
```

Subtract one from the WolframIndex (see

below) to get the length of line to use. So this is a period 87381 oscillator, length 36, WolframIndex of 37:

Code: Select all

```
x = 36, y = 1, rule = B3i4it5ry6k7e/S1e2k3ey4it5i
36o!
#C [[ AUTOSTART STEP 50 STOP 87381 ]]
```

Here are a few more of the diagonal-line periods tabulated. There seems to be a perfect match with the other mechanisms so far.

Code: Select all

```
B2-a3-i/S01c diagonal lines:
X = pattern dies out completely
Length Odd Even
------ ---- ----
1 (stable)
2 X
3 6
4 4
5 14
6 X
7 14
8 12
9 62
10 8
11 126
12 28
13 30
14 X
15 30
16 28
17 1022
18 24
19 126
20 124
21 4094
22 16
23 2046
24 252
25 1022
26 56
27 32766
28 60
29 62
30 X
31 62
32 60
33 8190
34 56
35 174762*
36 2044
37 8190
38 48
39 2046
40 252
41 254
42 248
```

The "Odd" column is mostly understandable. It's very close to

A160657. As Nathaniel points out, the number is usually two less than a power of two -- and the exponent is something called the "multiplicative suborder" (see the

A003558 sequence).

The "Even" column period for a diagonal line of length 2N+2, for N>1, seems to be always twice the period of a line of length N... unless the length is two less than a power of two, in which case all cells die. Which is just what El'endia Starman from the Quest for Tetris project says in

https://oeis.org/A268754: "For odd-indexed terms, a(2n+1) = 2*a(n), except when n is of the form (2^k - 1), in which case a(n) = 1."

There are occasional exceptions in the Odd column, where an oscillator returns to its original state too soon. The A160657 calculation does still give a number where the line returns to its original configuration -- but in just a few cases, that number is some multiple of the period, rather than the actual period.

In the weird cases so far, the multiple is always three. But a 2^N-2 period can perfectly well be divisible by three without displaying this subperiod behavior.

Example: the multiplicative suborder calculation gives the value 126 for a length-11 diagonal line. 126 is the actual period, not 126/3 = 42.

However, the multiplicative suborder calculation gives the value 524286 for a length-35 diagonal line. But the actual oscillator period is only a third of that -- 174762.