OFFSET
1,1
COMMENTS
The sequence A000984 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.
The number of n-cycles in the graph of overlapping m-permutations where n <= m. - Richard Ehrenborg, Dec 10 2013
a(n) is divisible by n (cf. A268619), 6*a(n) is divisible by n^2 (cf. A268592). - Max Alekseyev, Feb 09 2016
Apparently the number of Lyndon words of length n with a 4-letter alphabet (see A027377) where the first letter of the alphabet appears with the same frequency as the second of the alphabet. E.g a(1)=2 counts the words (2), (3), a(2)= 2 counts (01) (23), a(3)=6 counts (021) (031) (012) (013) (223) (233). R. J. Mathar, Nov 04 2021
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..1669
R. Ehrenborg, S. Kitaev and E. Steingrimsson, Number of cycles in the graph of 312-avoiding permutations, arXiv:1310.1520 [math.CO], 2013.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
FORMULA
a(n) = 2*A022553(n).
a(n) = A007727(n)/n. - R. J. Mathar, Jul 24 2017
G.f.: 2 * Sum_{k>=1} mu(k)*log((1 - sqrt(1 - 4*x^k))/(2*x^k))/k. - Ilya Gutkovskiy, May 18 2019
a(n) ~ 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 04 2022
EXAMPLE
a(5) = 50 because if a map has A000984 as its periodic points, then it would have 2 fixed points and 252 points of period 5, hence 50 orbits of length 5.
MAPLE
with(numtheory):
a:= n-> add(mobius(n/d)*binomial(2*d, d), d=divisors(n))/n:
seq(a(n), n=1..30); # Alois P. Heinz, Dec 10 2013
MATHEMATICA
a[n_] := (1/n)*Sum[MoebiusMu[d]*Binomial[2*n/d, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jul 16 2015 *)
PROG
(PARI) a(n)=sumdiv(n, d, moebius(n/d)*binomial(2*d, d))/n \\ Charles R Greathouse IV, Dec 10 2013
(Python)
from sympy import mobius, binomial, divisors
def a(n): return sum(mobius(n//d) * binomial(2*d, d) for d in divisors(n))//n
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Jul 24 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Thomas Ward, Mar 13 2001
STATUS
approved