A215331 Smooth necklaces with 5 colors.

1, 5, 9, 16, 35, 76, 190, 455, 1156, 2911, 7438, 18992, 48902, 125968, 325975, 845202, 2197690, 5725854, 14951308, 39110371, 102490649, 269002564, 707096093, 1861183847, 4905172383, 12942843424

Offset: 0

Author

Joerg Arndt, Aug 08 2012

Keywords

• nonn
• more

Comments

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

Examples

			The smooth pre-necklaces, necklaces (N), and Lyndon words (L) of length 4 with 4 colors (using symbols ".", "1", "2", "3", and "4") 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
    1234   4    1234  N L
    2222   1       2  N
    2223   4    2223  N L
    2232   3     232
    2233   4    2233  N L
    2234   4    2234  N L
    2323   2      23  N
    2332   3     332
    2333   4    2333  N L
    2334   4    2334  N L
    2343   4    2343  N L
    2344   4    2344  N L
    3333   1       3  N
    3334   4    3334  N L
    3343   3     343
    3344   4    3344  N L
    3434   2      34  N
    3443   3     443
    3444   4    3444  N L
    4444   1       4  N
There are 43 pre-necklaces, 35 necklaces, and 26 Lyndon words.
So a(4) = 35.
		

Crossrefs

Cf. A215327 (smooth necklaces, 3 colors) A215328 (smooth Lyndon words, 3 colors).

Extensions

More terms from Joerg Arndt, Jun 17 2019