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 A226893 Number of binary Lyndon words of length n having a conjugate at Hamming distance 2. 0

%I

%S 0,1,2,3,6,9,14,20,26,40,42,72,62,123,102,208,114,405,146,668,290,

%T 1201,222,2504,316,4356,800,8580,366,17866,422,33312,2514,66010,990,

%U 135960,614,262749,8870,526256,762,1066044,842

%N Number of binary Lyndon words of length n having a conjugate at Hamming distance 2.

%C A Lyndon word is an aperiodic word that is lexicographically least in its conjugacy class (of cyclic shifts). A conjugate is a cyclic shift. Hamming distance is the number of positions in which two strings differ.

%e a(7) = 14, because there are 18 binary Lyndon words of length 7, and all but the following 4 have a conjugate at Hamming distance 2: 0001011, 0001101, 0010111, 0011101.

%e From _Joerg Arndt_, Jun 21 2013: (Start)

%e There are a(8) = 20 Lyndon words of length 8 that have a conjugate at Hamming distance 2 (dots for zeros):

%e ct: word conjugate word XOR conjugate

%e 01: 1....... .1...... 11......

%e 02: 11...... .11..... 1.1.....

%e 03: 1.1..... ..1.1... 1...1...

%e 04: 111..... .111.... 1..1....

%e 05: 1..1.... ...1..1. 1.....1.

%e 06: 1111.... .1111... 1...1...

%e 07: 11..1... 1...11.. .1...1..

%e 08: 1.1.1... ..1.1.1. 1.....1.

%e 09: 1..11... 1...1..1 ...1...1

%e 10: 11111... .11111.. 1....1..

%e 11: 1.1..1.. 1..1.1.. ..11....

%e 12: 1.11.1.. 1..1.11. ..1...1.

%e 13: 111.11.. 11..111. ..1...1.

%e 14: 11.111.. 11..11.1 ...1...1

%e 15: 111111.. .111111. 1.....1.

%e 16: 111.1.1. 1.111.1. .1.1....

%e 17: 11.11.1. .1.11.11 1......1

%e 18: 11111.1. 1.11111. .1...1..

%e 19: 1111.11. 11.1111. ..1.1...

%e 20: 1111111. .1111111 1......1

%e (End)

%Y Cf. A001037.

%K nonn,more

%O 1,3

%A _Jeffrey Shallit_, Jun 21 2013

%E More terms from _Joerg Arndt_, Jun 21 2013 and Jun 26 2013

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Last modified September 20 18:26 EDT 2018. Contains 315240 sequences. (Running on oeis4.)