A306517 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).

1010100, 101000100, 101001000100, 10101001010100, 101001001000100, 101010010110100, 101110000000100, 1010100010110100, 1010101010010100, 10101000101110100, 10101001110110100, 101000100101000100, 101001001001000100, 101010000101110100, 101010010100110100

Offset: 1

Author

A.H.M. Smeets, Feb 21 2019

Keywords

• nonn
• base

Comments

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.

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..1701