OFFSET
1,2
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.
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. [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.
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).
EXAMPLE
a(5)=6 because we have six binary cyclic sequences of length 5 that avoid four consecutive zeros: 00011, 00101, 00111, 01101, 01111, 11111.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, Dec 29 2016
STATUS
approved