If the decimal representation of the binary string 10s00 is in the sequence, so is 101s000.
For binary representation see
A306515.
This sequence is a subset of
A066059.
These regular patterns can be represented 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,
where T_a0, T_b0, T_c0 and T_d0 are some terminating strings.
Numbers in this sequence are generated by choosing S_a or S_c from the starting symbol S.
The decimal representation of all binary numbers derived by S -> S_a -> 10 T_a 00 -> 10 T_a0 00 are given in sequence
A306516, its binary representation in
A306517.
Observed: all values are in the ranges lower(k)..upper(k), where lower(k) = 81*2^k + 2^floor((k+6)/2) + 2^6*(2^floor((k-8)/2) - 1) + 4, which holds for k >= 11, and upper(k) = 3*2^floor((k+4)/2)*(2^floor((k+7)/2) - 1), which holds for k >= 0; the number of terms in each successive range increases by about a factor of 4/3. All terms between lower(k) and upper(k) are represented by a (k+7)-binary-digit number (see
A306515). Each m-binary-digit number will have a successive number of m+1 binary digits in the next range. About 1/4 of each obtained number in this sequence has a new unique cyclic trajectory (see
A306516 and
A306517), i.e., a cyclic trajectory not joining a previous cyclic trajectory, which explains the growth factor of 4/3 for each successive range.
All terms
A061561(4k+2) for k >= 0 are included in this sequence.
All values in
A103897(k+3) for k >= 0 are included in this sequence.
