Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration

The rule 150 cellular automaton is a remarkable discrete dynamical system, as it shows 1 fα spectra if started from a single seed [J. Nagler and J. C. Claussen, Phys. Rev. E 71, 067103 (2005)]. Despite its simplicity, a feasible solution for its time behavior is not obvious. Its self-similarity does not follow a one-step iteration like other elementary cellular automata. Here it is shown how its time behavior can be solved as a two-step vectorial, or string, iteration, which can be viewed as a generalization of Fibonacci iteration generating the time series from a sequence of vectors of increasing length. This allows us to compute the total activity time series more efficiently than by simulating the whole spatiotemporal process or even by using the closed expression. The results are further extended to the generalization of rule 150 to the two-dimensional case and to Bethe lattices and the relation to corresponding integer sequences is discussed.