A274020 - OEIS

%I #44 Jun 24 2018 16:03:41

%S 1,5,15,44,160,604,2510,10545,45825,201669,900307,4057625,18447565,

%T 84444000,388878560,1799985435,8368841895,39062428790,182961584260,

%U 859612223990,4049955449888,19128675877279,90553562670495,429560546547595,2041573370075675,9719864998575489,46350124359578975,221352533355568044

%N Number of n-bead 5-ary 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 P. Hadjicostas and L. Zhang, <a href="https://doi.org/10.1016/j.disc.2018.03.007">On cyclic strings avoiding a pattern"</a>, Discrete Mathematics, 341 (2018), 1662-1674.

%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=5 is the number of symbols in the alphabet we are using. - _Petros Hadjicostas_, Sep 12 2017

%F Define sequence (c(n): n>=1) by c(1) = q, c(2) = q^2, c(3) = q^3-3, and c(n) = q*c(n-1) - c(n-3) for n>=4. Then a(n) = (1/n)*Sum_{d|n} phi(n/d)*c(d) for n>=1. (Here q=5.) - _Petros Hadjicostas_, Jan 29 2018

%e The following necklace:

%e .    1-1

%e .   /   \

%e .  0     0

%e .  |     |

%e .  1     3

%e .   \   /

%e .    2-4

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

%Y Cf. A000031, A274017, A274018, A274019.

%K nonn

%O 0,2

%A _Marko Riedel_, Jun 06 2016