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A274019 Number of n-bead quaternary necklaces (no turning over allowed) that avoid the subsequence 110. 3
1, 4, 10, 23, 66, 192, 636, 2092, 7228, 25175, 89212, 318808, 1150444, 4177908, 15268494, 56078527, 206903020, 766342160, 2848351388, 10619472284, 39702648534, 148806583111, 558999381656, 2104255629608, 7936108068008, 29982733437844, 113456750715426, 429964269551767, 1631663320986086 (list; graph; refs; listen; history; text; internal format)
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).

Because A(x) = Sum_{n>=1} a(n)*x^n = 1 - Sum_{n>=1} (phi(n)/n)*log(1-B(x^n)), where B(x) = q*x - x^3 and q = 4, we may find sequence (c(n): n>=1) that satisfies a(n) = (1/n)*Sum_{d|n} phi(n/d)*c(d) for n>=1 by using the formula Sum_{n>=1} c(n)*x^n = C(x) = x*(dB/dx)/(1-B(x)). In our case, C(x) = x*(d(q*x-x^3)/dx)/(1-(q*x-x^3)) = (q*x - 3*x^3)/(1 - q*x + x^3). This implies that 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. This comment applies not only to this sequence, but also to sequences A274017, A274018 and A274020 as well (corresponding to cases q=2, 3, and 5, respectively). - Petros Hadjicostas, Jan 31 2018

LINKS

Table of n, a(n) for n=0..28.

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=4 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=4.) - Petros Hadjicostas, Jan 29 2018

EXAMPLE

The following necklace

.   1-1

.  /   \

. 0     0

. |     |

. 1     3

.  \   /

.   0-2

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

CROSSREFS

Cf. A000031, A274017, A274018, A274020.

Sequence in context: A137531 A102549 A277789 * A008258 A008251 A174934

Adjacent sequences:  A274016 A274017 A274018 * A274020 A274021 A274022

KEYWORD

nonn

AUTHOR

Marko Riedel, Jun 06 2016

STATUS

approved

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Last modified May 17 16:03 EDT 2021. Contains 343980 sequences. (Running on oeis4.)