A215327 Smooth necklaces with 3 colors.

1, 3, 5, 8, 15, 27, 58, 115, 252, 541, 1196, 2629, 5894, 13156, 29667, 66978, 151966, 345497, 788396, 1802678, 4133161, 9495317, 21861393, 50423468, 116514553, 269666605, 625108573, 1451128479, 3373267275, 7851415838, 18296568717

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.

Links

Table of n, a(n) for n=0..30.

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

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

Sequence in context: A193147 A052977 A191633 * A208723 A285010 A352917

Adjacent sequences: A215324 A215325 A215326 * A215328 A215329 A215330

Keyword

nonn

Author

Joerg Arndt, Aug 08 2012

Extensions

More terms from Joerg Arndt, Jun 17 2019

Status

approved