|
%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
|
