A280218 Number of binary necklaces of length n with no subsequence 0000.

1, 2, 3, 5, 6, 11, 15, 27, 43, 75, 125, 228, 391, 707, 1262, 2285, 4119, 7525, 13691, 25111, 46033, 84740, 156123, 288529, 533670, 989305, 1835983, 3412885, 6351031, 11834623, 22074821, 41222028, 77048131, 144148859, 269913278, 505826391, 948652695, 1780473001, 3343960175, 6284560319, 11818395345

Offset: 1

Author

Petros Hadjicostas, Dec 29 2016

Keywords

• nonn

Comments

a(n) is the number of cyclic sequences of length n consisting of zeros and ones that do not contain four consecutive zeros provided we consider as equivalent those sequences that are cyclic shifts of each other.

Examples

			a(5)=6 because we have six binary cyclic sequences of length 5 that avoid four consecutive zeros: 00011, 00101, 00111, 01101, 01111, 11111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3521
• P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
• P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
• Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 57.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

Crossrefs

Row 4 of A322057.

Cf. A000358, A073817, A093305, A280303.

Programs

Mathematica

Table[(1/n) Sum[EulerPhi[n/d] SeriesCoefficient[(4 - 3 x - 2 x^2 - x^3)/(1 - x - x^2 - x^3 - x^4), {x, 0, d}], {d, Divisors@ n}], {n, 41}] (* Michael De Vlieger, Dec 30 2016 *)

PARI

N=44; x='x+O('x^N);
B(x)=x*(1+x+x^2+x^3);
Vec(sum(k=1, N, eulerphi(k)/k * log(1/(1-B(x^k))))) \\ Joerg Arndt, Dec 29 2016

Formula

a(n) = (1/n) * Sum_{d divides n} totient(n/d) * A073817(d).

G.f.: Sum_{k>=1} (phi(k)/k) * log(1/(1-B(x^k))) where B(x) = x*(1+x+x^2+x^3).