Many regularities are apparent in Table 1. For example, it is easy to spot numerous occurences of , , , , etc., and even to guess the general pattern of such occurences. Such results can usually be proved by the method of [3]. That is, one guesses the expansion of , verifies that this purported expansion is a legitimate expansion of a beta number according to Parry's criterion, computes the characteristic polynomial of by (1.2), and verifies that is divisible by , the minimal polynomial of . For example, here are some simple cases: