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

[hv at crypt.org]

Thanks, A222187 will also of course be treating the 2 x 2 cases 11 / 00
and 10 / 10 as distinct, hence a(2) = 7.

The linked A222188 is what I was really after though, to try and consider
if I can come up with any remotely efficient algorithm to enumerate
the distinct cases.

Hugo

Allan Wechsler <acwacw at gmail.com> wrote:
: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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