A215329 Smooth necklaces with 4 colors.

1, 4, 7, 12, 25, 51, 121, 272, 656, 1563, 3794, 9193, 22529, 55189, 136025, 335942, 832605, 2068070, 5150558, 12852754, 32139908, 80509629, 202016993, 507669052, 1277595853, 3219366640, 8122296152

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.

Links

Table of n, a(n) for n=0..26.

Example

The smooth pre-necklaces, necklaces (N), and Lyndon words (L) of length 4 with 4 colors (using symbols ".", "1", "2", and "3") 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
    .123   4    .123  N L
    1111   1       1  N
    1112   4    1112  N L
    1121   3     121
    1122   4    1122  N L
    1123   4    1123  N L
    1212   2      12  N
    1221   3     221
    1222   4    1222  N L
    1223   4    1223  N L
    1232   4    1232  N L
    1233   4    1233  N L
    2222   1       2  N
    2223   4    2223  N L
    2232   3     232
    2233   4    2233  N L
    2323   2      23  N
    2332   3     332
    2333   4    2333  N L
    3333   1       3  N
There are 31 pre-necklaces, 25 necklaces, and 18 Lyndon words.
So a(4) = 25.

Crossrefs

Cf. A215327 (smooth necklaces, 3 colors)

Sequence in context: A243860 A322619 A299900 * A208724 A183336 A102953

Adjacent sequences: A215326 A215327 A215328 * A215330 A215331 A215332

Keyword

nonn,more

Author

Joerg Arndt, Aug 08 2012

Extensions

More terms from Joerg Arndt, Jun 17 2019

Status

approved