OFFSET
0,2
COMMENTS
We call a necklace (x[1],x[2],...,x[n]) smooth if abs(x[k]-x[k-1]) <= 1 for 2<=k<=n.
All binary necklaces (2 colors, A000031) are necessarily smooth.
EXAMPLE
The smooth pre-necklaces, necklaces (N), and Lyndon words (L) of length 4 with 3 colors (using symbols ".", "1", and "2") are:
.... 1 . N
...1 4 ...1 N L
..1. 3 .1.
..11 4 ..11 N L
..12 4 ..12 N L
.1.1 2 .1 N
.11. 3 11.
.111 4 .111 N L
.112 4 .112 N L
.121 4 .121 N L
.122 4 .122 N L
1111 1 1 N
1112 4 1112 N L
1121 3 121
1122 4 1122 N L
1212 2 12 N
1221 3 221
1222 4 1222 N L
2222 1 2 N
There are 19 pre-necklaces, 15 necklaces, and 10 Lyndon words.
So a(4) = 15.
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt, Aug 08 2012
EXTENSIONS
More terms from Joerg Arndt, Jun 17 2019
STATUS
approved