Mention of the Thue-Morse and `Rabbit' sequences gives me the opportunity to rabbit on about some private passions. Both are examples of a class called "D0L", characterised as fixed points of some context-free morphism or "inflation rule", which is just one of many ways in which the self-similarity remarked on might be formalised. I prefer to discuss what I christen "D0LEC" sequences, in an attempt to define a complexity class above periodic and linear recurring sequences (over a finite domain), and including such things as automatic or characteristic or 2-distance sequences. A D0LEC sequence is defined by two morphisms (maps from symbols to strings), both having constant length of RHS string: an "inflation" morphism generating the fixed point sequence via iteration, supplemented by an "extension" morphism translating that into another final sequence via a single application. Example: To define the Thue-Morse sequence, instead of using the elegant morphism 0 -> 01, 1 -> 10 we could be particularly perverse and define it by the inflation f: A -> BC, B -> BD, C -> CA, D -> CB, generating the one-sided sequence on 4 symbols (starting from B) V_n = BDCBCABD CABCBDCB CABCBDCA BDCBCABD ... , followed by the binary extension A -> 0, B -> 0, C -> 1, D -> 1, so generating the usual binary Thue-Morse T_n = 01101001 10010110 10010110 01101001 ... ; or alternatively, we might apply the ternary extension A -> 0,\ B -> 1,\ C -> 2,\ D -> 0, so generating the ternary Thue-Morse U_n = 10212010 20121021 20121020 10212012 ... , in which successive pairs of binary digits have been re-coded as single ternary digits U_n = (2 T_n + T_{n+1}) mod 3. Why bother? It is well-known that U_n is "square-free", having no (nonempty) block of digits followed immediately by a copy of itself; and that T_n is "cube-free", having no block followed by two copies. The proofs of these facts traditionally seem to involve an awful lot of tricky chasing down alternative branches; I claim that in contrast, it is (relatively) easy to show that the quaternary sequence V_n is square-free, whence the results for U_n, T_n follow straightforwardly. But we can do more ... By inspection, it is easily established that no infinite binary sequence can be square-free: for example, T_n contains the blocks 00, 11, 10011001, 01100110, ... However, we might reasonably enquire how closely a binary sequence may approach this arcadian state. A sequence will be defined "k-square-free" if no block of length exceeding k/2 is squared: for example, the finite string 010011000111001101 is 4-square-free. So now I ask: What is the smallest k for which some k-square-free binary sequence exists? Fred Lunnon, Maynooth Dec 2005