Neil J. A. Sloane, Rutgers Experimental Mathematics Seminar, February 5, 2015
See part 2 at vimeo.com/119073819
Abstract: A cellular automaton (or CA) is started with a single ON cell; how many cells are ON after n generations? A general theorem will be presented which applies to a certain class of "odd-rule" CAs, including Rule 150, Rule 614, and Fredkin's Replicator, although to get an explicit answer in the last two requires delicate surgical techniques. A number of other CAs can be analyzed by ad hoc methods, although most two-dimensional CAs seem beyond reach. The difficulty of analysis is strongly correlated with the beauty of the resulting patterns.