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

0,2

Comments

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

Links

Table of n, a(n) for n=0..27.

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.

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

Example

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.

Crossrefs

Cf. A000031, A274017, A274018, A274019.

Sequence in context: A064453 A059251 A295179 * A109952 A146632 A076103

Adjacent sequences: A274017 A274018 A274019 * A274021 A274022 A274023

Keyword

nonn

Author

Marko Riedel, Jun 06 2016

Status

approved