Periods in rule 37R

With a system of size n, the maximum possible repetition period is 2^{2n}. In actuality, however, the periods are considerably shorter. With all cells 0 on one step, and a block of nonzero cells on the next step, the periods are for example: {1}: 21; {1, 1}: 3n - 8; {1, 0, 1}: 666; {1, 1, 1}: 3n - 8; {1, 0, 0, 1}: irregular (< 24n; peaks at 6j + 1); {1, 0, 0, 1, 0, 1}: irregular (≲ 2^{n}; 857727 for n= 2 6; 13705406 for n = 100). With completely random initial conditions, there are great fluctuations, but a typical period is around 2^{n/3}.