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A215331 Smooth necklaces with 5 colors.

%I A215331 #12 Jun 17 2019 05:46:55
%S A215331 1,5,9,16,35,76,190,455,1156,2911,7438,18992,48902,125968,325975,
%T A215331 845202,2197690,5725854,14951308,39110371,102490649,269002564,
%U A215331 707096093,1861183847,4905172383,12942843424
%N A215331 Smooth necklaces with 5 colors.
%C A215331 We call a necklace (x[1],x[2],...,x[n]) smooth if abs(x[k]-x[k-1]) <= 1 for 2<=k<=n.
%e A215331 The smooth pre-necklaces, necklaces (N), and Lyndon words (L) of length 4 with 4 colors (using symbols ".", "1", "2", "3", and "4") are:
%e A215331     ....   1       .  N
%e A215331     ...1   4    ...1  N L
%e A215331     ..1.   3     .1.
%e A215331     ..11   4    ..11  N L
%e A215331     ..12   4    ..12  N L
%e A215331     .1.1   2      .1  N
%e A215331     .11.   3     11.
%e A215331     .111   4    .111  N L
%e A215331     .112   4    .112  N L
%e A215331     .121   4    .121  N L
%e A215331     .122   4    .122  N L
%e A215331     .123   4    .123  N L
%e A215331     1111   1       1  N
%e A215331     1112   4    1112  N L
%e A215331     1121   3     121
%e A215331     1122   4    1122  N L
%e A215331     1123   4    1123  N L
%e A215331     1212   2      12  N
%e A215331     1221   3     221
%e A215331     1222   4    1222  N L
%e A215331     1223   4    1223  N L
%e A215331     1232   4    1232  N L
%e A215331     1233   4    1233  N L
%e A215331     1234   4    1234  N L
%e A215331     2222   1       2  N
%e A215331     2223   4    2223  N L
%e A215331     2232   3     232
%e A215331     2233   4    2233  N L
%e A215331     2234   4    2234  N L
%e A215331     2323   2      23  N
%e A215331     2332   3     332
%e A215331     2333   4    2333  N L
%e A215331     2334   4    2334  N L
%e A215331     2343   4    2343  N L
%e A215331     2344   4    2344  N L
%e A215331     3333   1       3  N
%e A215331     3334   4    3334  N L
%e A215331     3343   3     343
%e A215331     3344   4    3344  N L
%e A215331     3434   2      34  N
%e A215331     3443   3     443
%e A215331     3444   4    3444  N L
%e A215331     4444   1       4  N
%e A215331 There are 43 pre-necklaces, 35 necklaces, and 26 Lyndon words.
%e A215331 So a(4) = 35.
%Y A215331 Cf. A215327 (smooth necklaces, 3 colors) A215328 (smooth Lyndon words, 3 colors).
%K A215331 nonn,more
%O A215331 0,2
%A A215331 _Joerg Arndt_, Aug 08 2012
%E A215331 More terms from _Joerg Arndt_, Jun 17 2019