A215327 - OEIS

%I #14 Jun 17 2019 05:36:32

%S 1,3,5,8,15,27,58,115,252,541,1196,2629,5894,13156,29667,66978,151966,

%T 345497,788396,1802678,4133161,9495317,21861393,50423468,116514553,

%U 269666605,625108573,1451128479,3373267275,7851415838,18296568717

%N Smooth necklaces with 3 colors.

%C We call a necklace (x[1],x[2],...,x[n]) smooth if abs(x[k]-x[k-1]) <= 1 for 2<=k<=n.

%C All binary necklaces (2 colors, A000031) are necessarily smooth.

%e The smooth pre-necklaces, necklaces (N), and Lyndon words (L) of length 4 with 3 colors (using symbols ".", "1", and "2") are:

%e     ....   1       .  N

%e     ...1   4    ...1  N L

%e     ..1.   3     .1.

%e     ..11   4    ..11  N L

%e     ..12   4    ..12  N L

%e     .1.1   2      .1  N

%e     .11.   3     11.

%e     .111   4    .111  N L

%e     .112   4    .112  N L

%e     .121   4    .121  N L

%e     .122   4    .122  N L

%e     1111   1       1  N

%e     1112   4    1112  N L

%e     1121   3     121

%e     1122   4    1122  N L

%e     1212   2      12  N

%e     1221   3     221

%e     1222   4    1222  N L

%e     2222   1       2  N

%e There are 19 pre-necklaces, 15 necklaces, and 10 Lyndon words.

%e So a(4) = 15.

%Y Cf. A001867 (necklaces, 3 colors), A215328 (smooth Lyndon words, 3 colors).

%K nonn

%O 0,2

%A _Joerg Arndt_, Aug 08 2012

%E More terms from _Joerg Arndt_, Jun 17 2019