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
KEYWORD
easy,nonn
AUTHOR
Philippe Deléham, Apr 24 2004
STATUS
approved