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

[Allan Wechsler]

This is a job for the Frobenius-Burnside-Cauchy-Polya counting formula. I
am sure this sequence is in OEIS.

[pauses for 15 minutes to work it out]

OK, after working out F(4) = 34 by hand, I found A222187, "Number of
toroidal n X 2 binary arrays ...", which counts the number of cases we have
to try to find the longest-lived Life pattern on such an array. It doesn't
tell us how to enumerate them, though.



On Fri, May 31, 2019 at 8:21 PM <hv at crypt.org> wrote:

> 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.
>
> I've started thinking about this, and am unsure if I've confused myself.
>
> 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
>
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