A060168 Number of orbits of length n under the map whose periodic points are counted by A001643.

1, 1, 1, 2, 4, 6, 10, 15, 26, 42, 74, 121, 212, 357, 620, 1064, 1856, 3209, 5618, 9794, 17192, 30153, 53114, 93554, 165308, 292250, 517802, 918207, 1630932, 2899434, 5161442, 9196168, 16402764, 29281168, 52319364, 93555601, 167427844, 299841117, 537357892, 963641588, 1729192432

Offset

1,4

Comments

The sequence A001643 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..41.

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 divides n } mu(d)*A001643(n/d).

Example

u(7) = 10 since a map whose periodic points are counted by A001643 would have 1 fixed point and 71 points of period 7, hence 10 orbits of length 7.

Prog

(PARI) a001643(n) = if(n<0, 0, polcoeff(x*(1+2*x+4*x^3+5*x^4+6*x^5)/(1-x-x^2-x^4-x^5-x^6)+x*O(x^n), n))
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001643(n/d)); \\ Michel Marcus, Sep 11 2017

Crossrefs

Cf. A001642, A060164, A060165, A060166, A060167, A060169, A060170, A060171, A060171.

Sequence in context: A120549 A167270 A355108 * A113117 A179531 A134682

Adjacent sequences: A060165 A060166 A060167 * A060169 A060170 A060171

Keyword

easy,nonn

Author

Thomas Ward, Mar 13 2001

Extensions

More terms from Michel Marcus, Sep 11 2017

Status

approved