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 contextfree 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.
