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%I #23 May 07 2019 18:34:01
%S 1010100,101000100,101001000100,10101001010100,101001001000100,
%T 101010010110100,101110000000100,1010100010110100,1010101010010100,
%U 10101000101110100,10101001110110100,101000100101000100,101001001001000100,101010000101110100,101010010100110100
%N a(n) is the binary number with string structure 10s00, s in {0,1}*, such that it results in a non-palindromic cycle of length 4 in the Reverse and Add! procedure in base 2 and is not in the trajectory of any m < a(n).
%C 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:
%C S -> S_a | S_b | S_c | S_d
%C S_a -> 10 T_a 00, T_a -> 1 T_a 0 | T_a0;
%C S_b -> 11 T_b 01, T_b -> 0 T_b 1 | T_b0;
%C S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0;
%C S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0;
%C T_a0, T_b0, T_c0 and T_d0 being some terminating strings.
%C The strings obtained by S -> S_a -> 10 T_a0 00 are also called (the representation of) "cycle seeds".
%C 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.
%H A.H.M. Smeets, <a href="/A306517/b306517.txt">Table of n, a(n) for n = 1..1701</a>
%K nonn,base
%O 1,1
%A _A.H.M. Smeets_, Feb 21 2019
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