A060171 Number of orbits of length n under a map whose periodic points seem to be counted by A006953.

12, 54, 80, 30, 24, 5400, 0, 990, 1568, 636, 24, 2720, 0, 240, 5704, 510, 0, 3835776, 0, 26724, 3600, 108, 24, 89760, 0, 240, 1064, 120, 24, 113569300, 0, 510, 11752, 0, 264, 278281640

Offset

1,1

Comments

The sequence A006953 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.

Links

Table of n, a(n) for n=1..36.

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)*A006953(n/d).

Example

u(3) = 80 since a map whose periodic points are counted by A006953 has 12 fixed points and 252 points of period 3, hence 80 orbits of length 3.

Prog

(PARI) a(n) = sumdiv(n, d, moebius(d)*denominator(bernfrac(2*n/d)/(2*n/d)))/n; \\ Michel Marcus, Sep 10 2017

Crossrefs

Cf. A006953, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060172, A060173.

Sequence in context: A211060 A242514 A030182 * A133078 A034436 A186210

Adjacent sequences: A060168 A060169 A060170 * A060172 A060173 A060174

Keyword

easy,nonn

Author

Thomas Ward, Mar 13 2001

Status

approved