Sequences of preimages in elementary cellular automata

We search for regularities in the sequences of numbers of preimages for elementary cellular automata. For 46 out of 88 "minimal" rules, we find recognizable patterns, usually in the form of second order recurrence equations with constant coefficients. Introducing the concept of asymptotic emulation of CA rules, we then show how the regularities in the sequences of preimage numbers can be used to find rules emulating identity. We also show that the average density of nonzero sites after arbitrary number of steps (starting from disordered configuration) can be computed using the sequences of preimage numbers.