A093305 Number of binary necklaces of length n with no subsequence 000.

1, 2, 3, 4, 5, 9, 11, 19, 29, 48, 75, 132, 213, 369, 627, 1083, 1857, 3244, 5619, 9844, 17205, 30229, 53115, 93701, 165313, 292464, 517831, 918578, 1630933, 2900109, 5161443, 9197251, 16402841, 29283026, 52319379, 93558968, 167427845, 299846737, 537358107, 963651447, 1729192433

Offset

1,2

References

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 500.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..500

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.

Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), #16.8.2.

Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 57.

L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

Formula

a(n) = (1/n) * Sum_{d divides n} totient(n/d)*A001644(d).

G.f.: Sum_{k>=1} phi(k)/k * log( 1/(1-B(x^k)) ) where B(x) = x*(1+x+x^2). - Joerg Arndt, Aug 06 2012

a(n) ~ d^n / n, where d = (19 + 3*sqrt(33))^(1/3)/3 + 4/(3*(19 + 3*sqrt(33))^(1/3)) + 1/3 = A058265 = 1.8392867552141611325518... - Vaclav Kotesovec, Jul 13 2019

Mathematica

Table[1/n * Sum[EulerPhi[n/d] (d Sum[Sum[Binomial[j, d - 3 k + 2 j] Binomial[k, j], {j, d - 3 k, k}]/k, {k, d}]), {d, Divisors@ n}], {n, 41}] (* Michael De Vlieger, Dec 28 2016, after Vladimir Joseph Stephan Orlovsky at A001644 *)

Prog

(PARI)
N=66;  x='x+O('x^N);
B(x)=x*(1+x+x^2);
A=sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k))));
Vec(A)
/* Joerg Arndt, Aug 06 2012 */

Crossrefs

Row 3 of A322057.

Cf. A000358, A001644, A280218, A280303.

Sequence in context: A139441 A333612 A362025 * A065817 A361227 A084542

Adjacent sequences: A093302 A093303 A093304 * A093306 A093307 A093308

Keyword

easy,nonn

Author

Philippe Deléham, Apr 24 2004

Status

approved