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Number of n-bead ternary necklaces (no turning over allowed) that avoid the subsequence 110.
5

%I #43 Jun 24 2018 18:34:05

%S 1,3,6,10,21,42,103,237,603,1519,3942,10257,27131,71940,192462,516933,

%T 1395636,3781356,10283911,28050600,76732047,210414811,578330649,

%U 1592821005,4395261552,12149386569,33637309323,93267459520,258961863288,719938597227,2003881480452,5583818718102,15575529493713

%N Number of n-bead ternary necklaces (no turning over allowed) that avoid the subsequence 110.

%C The pattern in this enumeration must be contiguous (all three values next to each other in one sequence of three letters).

%H Math Stackexchange, Marko Riedel et al., <a href="http://math.stackexchange.com/questions/1812920/">Counting circular sequences</a>.

%H Marko Riedel, <a href="/A274017/a274017.maple.txt">Maple code for this sequence</a>.

%F G.f.: 1 - Sum_{n>=1} (phi(n)/n)*log(x^(3*n) - q*x^n + 1), where q=3 is the number of symbols in the alphabet we are using. - _Petros Hadjicostas_, Sep 12 2017

%F a(n) = (1/n)*Sum_{d|n} phi(n/d)*A215885(d) for n >= 1. - _Petros Hadjicostas_, Sep 13 2017

%e The necklace

%e 1--1

%e / \

%e 0 0

%e | |

%e 1 2

%e \ /

%e 0--0

%e contains one instance of the subsequence starting in the upper left corner. Unlike a bracelet, the necklace is oriented.

%Y Cf. A000031, A274017, A274019, A274020.

%K nonn

%O 0,2

%A _Marko Riedel_, Jun 06 2016