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A306515 Binary numbers 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. 3
1010100, 10110100, 101000100, 101101000, 1010101100, 1011101000, 10110001000, 10111010000, 101001000100, 101101011000, 101111010000, 1010111001100, 1011100010000, 1011110100000, 10101001010100, 10110010001000, 10111010110000, 10111110100000, 101001001000100, 101010010110100 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If the binary representation of the binary string 10s00 is in the sequence, so is 101s000.

For decimal representation see A306514.

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.

From the fact that all strings derived from S_b have prefix 11 and suffix 00, it can be proved that all strings derived from S_a must have prefix 111 (i.e., 1 is prefix of s, with s as in the name of this sequence). Similarly, from the fact that all strings derived from S_d have prefix 11 and suffix 000, it can be proved that all strings derived from S_c must have prefix 111 (i.e., again, 1 is prefix of s, with s as in the name of this sequence). In the later case, 11 is a prefix of s, which is even stronger. I believe additional stronger conditions can be observed and proved, so I challenge others to take a look at it too.

LINKS

A.H.M. Smeets, Table of n, a(n) for n = 1..6976

CROSSREFS

Cf. A306514, A306516, A306517.

Sequence in context: A066354 A220289 A133219 * A306517 A043643 A269564

Adjacent sequences:  A306512 A306513 A306514 * A306516 A306517 A306518

KEYWORD

nonn,base

AUTHOR

A.H.M. Smeets, Feb 21 2019

STATUS

approved

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Last modified February 27 10:31 EST 2020. Contains 332304 sequences. (Running on oeis4.)