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We discuss example of an elementary cellular automaton for which the densityof ones decays toward its limiting value as a power of the number of iterations$n$. Using the fact that this rule conserves the number of blocks 10 and thatpreimages of some other blocks exhibit patterns closely related to patternsobserved in rule 184, we derive expressions for the number of $n$-steppreimages of all blocks of length 3. These expressions involve Catalan numbers,and together with basic properties of iterated probability measures they allowus to to compute the density of ones after $n$ iterations, as well asprobabilities of occurrence of arbitrary block of length smaller or equal to 3.

Catalan numbers and power laws in cellular automaton rule 14