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Base-2 analog of A063048 (base 10) and A075421 (base 4); subsequence of A066059. - For the trajectory of 22 (cf. A061561) and the trajectory of 77 (cf. A075253) it has been proved that they do not contain a palindrome. A similar proof can be given for most terms of this sequence, but there are a few terms (4262, 17498, 33378, 33898, ...) whose trajectory does not show the kind of regularity that can be utilized for the construction of a proof. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few 'Reverse and Add!' steps (at most 84 for numbers < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.

From A.H.M. Smeets, Feb 12 2019: (Start)

Most terms in this sequence eventually give rise to a regular binary pattern. These regular patterns can be represented by contextfree grammars:

S_a -> 10 T_a 00, T_a -> 1 T_a 0 | A_a(n);

S_b -> 11 T_b 01, T_b -> 0 T_b 1 | B_a(n);

S_c -> 10 T_c 000, T_c -> 1 T_c 0 | C_a(n) and

S_d -> 11 T_d 101, T_d -> 0 T_d 1 | D_a(n).

A_22 = 1101, B_22 = 1000, C_22 = 1101, D_22 = 0010 (see also A058042);

A_77 = 1100010, B_77 = 0000101, C_77 = 1101011, D_77 = 0100000 (see also A075253)

Decimal representations for 10 A_a(n) 00 are given by A306514(n).

Binary representations for 10 A_a(n) 00 are given by A306515(n).

(End)

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.