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A093305 Number of binary necklaces of length n with no subsequence 000. 8
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 (list; graph; refs; listen; history; text; internal format)
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.

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

Cf. A000358, A001644, A280218, A280303.

Sequence in context: A100797 A139441 A333612 * A065817 A084542 A038201

Adjacent sequences:  A093302 A093303 A093304 * A093306 A093307 A093308

KEYWORD

easy,nonn

AUTHOR

Philippe Deléham, Apr 24 2004

STATUS

approved

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Last modified November 25 14:21 EST 2020. Contains 338624 sequences. (Running on oeis4.)