Rule 30

The left-hand side of the pattern shown has an obvious repetitive character. In general, if one looks along a diagonal n cells in from either edge of the pattern, then the period of repetition can be at most 2^{n}. On the right-hand edge, the first few periods that are seen are {1, 2, 2, 4, 8, 8, 16, 32, 32, 64, 64, 64, 64, 64, 128, 256} and in general the period seems to increase exponentially with depth. On the left-hand edge, the period increases only extremely slowly: period 2 is first achieved at depth 3, period 4 at depth 8, 8 at 29, 16 at 400, 32 at 87,867, 64 at 2,107,985,255 or more, and so on. (Each period doubling turns out to occur exactly when a diagonal in the pattern eventually becomes a white stripe, and the diagonal to its left has an odd number of black cells in each repeating block.) The boundary that separates repetition on the left from randomness on the right moves an average of about 0.252 cells to the left at every step (compare page 949). The picture below shows the fluctuations around this average.

Complete pattern.

All possible blocks appear to occur eventually (see page 725). The probability for a block of n adjacent white cells (corresponding to a row in a white triangle) seems quite accurately to approach 2^{-n}, with the first length 10 such block occurring at step 67 and the first length 20 one occurring at step 515.

Center column.

The pictures below show the excess of black over white cells in the center column. Out of the first 100,000 cells, a total of 50,098 are black, and out of the first million 500,768 are. The longest run of identical colors in the first 100,000 cells consists of 21 black cells, and in the first million elements 22 black cells. The first n elements can be found efficiently using

Module[{a = 1}, Table[First[IntegerDigits[a, a = BitXor[a, BitOr[2a, 4a]]; 2, i]], {i, n}]]

The sequence does not repeat in at least its first million steps, and I would amazed if it ever repeats, but as of now I know of no rigorous proof of this. (Erica Jen showed in 1986 that no pair of columns can ever repeat, and the arguments on page 1087 suggest that neither can the center column together with occasional neighboring cells.)