A274020 Number of n-bead 5-ary necklaces (no turning over allowed) that avoid the subsequence 110.

1, 5, 15, 44, 160, 604, 2510, 10545, 45825, 201669, 900307, 4057625, 18447565, 84444000, 388878560, 1799985435, 8368841895, 39062428790, 182961584260, 859612223990, 4049955449888, 19128675877279, 90553562670495, 429560546547595, 2041573370075675, 9719864998575489, 46350124359578975, 221352533355568044

Offset: 0

Author

Marko Riedel, Jun 06 2016

Keywords

• nonn

Comments

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

Examples

			The following necklace:
.    1-1
.   /   \
.  0     0
.  |     |
.  1     3
.   \   /
.    2-4
contains one instance of the subsequence starting in the upper left corner. Unlike a bracelet, the necklace is oriented.
		

Links

• P. Hadjicostas and L. Zhang, On cyclic strings avoiding a pattern", Discrete Mathematics, 341 (2018), 1662-1674.
• Math Stackexchange, Marko Riedel et al., Counting circular sequences.
• Marko Riedel, Maple code for this sequence.

Crossrefs

Cf. A000031, A274017, A274018, A274019.

Formula

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

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