A215329 - OEIS

%I #13 Jun 17 2019 07:31:04

%S 1,4,7,12,25,51,121,272,656,1563,3794,9193,22529,55189,136025,335942,

%T 832605,2068070,5150558,12852754,32139908,80509629,202016993,

%U 507669052,1277595853,3219366640,8122296152

%N Smooth necklaces with 4 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.

%e The smooth pre-necklaces, necklaces (N), and Lyndon words (L) of length 4 with 4 colors (using symbols ".", "1", "2", and "3") 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     .123   4    .123  N L

%e     1111   1       1  N

%e     1112   4    1112  N L

%e     1121   3     121

%e     1122   4    1122  N L

%e     1123   4    1123  N L

%e     1212   2      12  N

%e     1221   3     221

%e     1222   4    1222  N L

%e     1223   4    1223  N L

%e     1232   4    1232  N L

%e     1233   4    1233  N L

%e     2222   1       2  N

%e     2223   4    2223  N L

%e     2232   3     232

%e     2233   4    2233  N L

%e     2323   2      23  N

%e     2332   3     332

%e     2333   4    2333  N L

%e     3333   1       3  N

%e There are 31 pre-necklaces, 25 necklaces, and 18 Lyndon words.

%e So a(4) = 25.

%Y Cf. A215327 (smooth necklaces, 3 colors)

%K nonn,more

%O 0,2

%A _Joerg Arndt_, Aug 08 2012

%E More terms from _Joerg Arndt_, Jun 17 2019