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

[Antti Karttunen]

Cheers all,

here's a project that would require some computing power and/or time
and might offer nice insights for optimization:

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.

Then with "enough" terms, submit sequence where a(n) is the length of
that maximal cyclic sequence for nxn board. Of course there might be
several distinct maximal ones where none of the patterns are simply
rotations or reflections of any pattern of some other cyclic set.

My conjecture is that for n=8, a(n) = 132. See https://oeis.org/A179412
Clearly also a(2n) >= a(n), because 2n x 2n board can be divided into
four quadrants, and each quadrant initialized with the maximal
representative pattern of n x n board, thinking that it is the lone
occupant of its n x n toroidal universe.

If anybody starts coding this, then please let your program record
also all successive candidates for a maximal pattern of each n x n
board.



Best regards,

Antti Karttunen