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A060165 Number of orbits of length n under the map whose periodic points are counted by A000984. 16
2, 2, 6, 16, 50, 150, 490, 1600, 5400, 18450, 64130, 225264, 800046, 2865226, 10341150, 37566720, 137270954, 504171432, 1860277042, 6892317200, 25631327190, 95640829922, 357975249026, 1343650040256, 5056424257500, 19073789328750, 72108867614796 (list; graph; refs; listen; history; text; internal format)
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

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) = (1/n) * Sum_{d|n} mu(d) A000984(n/d) with mu = A008683.

a(n) = 2*A022553(n).

a(n) = A007727(n)/n. - R. J. Mathar, Jul 24 2017

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 map(a, xrange(1, 31)) # Indranil Ghosh, Jul 24 2017

CROSSREFS

Cf. A000984, A007727, A060164, A060166, A060167, A060168, A060169, A060170, A060171, A060172, A060173.

Sequence in context: A180068 A034439 A230825 * A134295 A184845 A062833

Adjacent sequences:  A060162 A060163 A060164 * A060166 A060167 A060168

KEYWORD

easy,nonn

AUTHOR

Thomas Ward (t.ward(AT)uea.ac.uk), Mar 13 2001

STATUS

approved

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Last modified October 19 13:38 EDT 2018. Contains 316361 sequences. (Running on oeis4.)