# [seqfan] Re: Finding the longest period Life-patterns on nxn toroidal board?

hv at crypt.org hv at crypt.org
Sat Jun 1 00:53:30 CEST 2019

```Antti Karttunen <antti.karttunen at gmail.com> wrote:
:Find the (representatives of) patterns that produce a maximal possible
:cyclic sequence of patterns on nxn toroidal board, with Conway's
:"Life" cellular automaton rules.

On a 1 x n board the number of distinguishable starting positions seems
to be A000029 (which makes sense). Richer results for Conway's Life
start with a 7-cycle at n=7 (1101000, a 3-place glider), and 6- and
8-cycles at n=8.

On a 2 x n board, the analagous sequence appears to start 1,3,6,13,33,74
which is not in the OEIS.

Have I miscalculated? If it needs to be added, what's a useful way to
describe the 2D analogue of a bracelet?

For reference, below are the 13 distinguishable results I find for n=3
on a 2xn grid.

Hugo

... (1)
...

x.. (6 analogues, including itself)
...

xx. (6)
...

x.. (3)
x..

x.. (6)
.x.

xxx (2)
...

xx. (12)
x..

xx. (6)
..x

xxx (6)
x..

xx. (3)
xx.

xx. (6)
x.x

xxx (6)
xx.

xxx (1)
xxx

total: 64 = 2^6 analagues accounted for.

END

```