A280303 Number of binary necklaces of length n with no subsequence 00000.

1, 2, 3, 5, 7, 12, 17, 31, 51, 91, 155, 287, 505, 930, 1695, 3129, 5759, 10724, 19913, 37239, 69643, 130745, 245715, 463099, 873705, 1651838, 3126707, 5927817, 11251031, 21382558, 40679233, 77475673, 147694719, 281822847, 538213671, 1028714071, 1967728553

Offset: 1

Author

Petros Hadjicostas and Lingyun Zhang, Dec 31 2016

Keywords

• nonn

Comments

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

Examples

			a(5)=7 because we have seven binary cyclic sequences (necklaces) of length 5 that avoid five consecutive zeros: 00001, 00011, 00101, 00111, 01101, 01111, 11111.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
• Petros Hadjicostas, Proof of the formula for the generating function from the formula for a(n)
• 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]
• L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

Crossrefs

Row 5 of A322057.

Cf. A000358, A093305, A280218, A074048.

Formula

a(n) = (1/n) * Sum_{d divides n} totient(n/d) * A074048(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+x^4).

Extensions

a(34) onwards from Andrew Howroyd, Jan 25 2024