It is conjectured that all terms in this sequence are derived from S -> S_a -> 10 T_a 00 -> 10 T_a0 00 when we represent the cyclic patterns by the context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | T_a0;
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | T_b0;
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0;
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0;
T_a0, T_b0, T_c0 and T_d0 being some terminating strings.
The strings obtained by S -> S_a -> 10 T_a0 00 are also called (the representation of) "cycle seeds".
It is observed that all strings 10 T_a0 00, with T_a0 produced by the extended right regular grammar with starting symbol T_a0 and production rules T_a0 -> 1010010 T_a0 | 101, are included in this sequence; i.e. the set of seeds in base 2 is infinite if this conjecture is true. Similar observations can be made for any base b = 2^a, a > 0.