A060167 Number of orbits of length n under the map whose periodic points are counted by A001642.

1, 1, 1, 2, 4, 5, 9, 13, 23, 36, 63, 101, 175, 290, 497, 840, 1445, 2460, 4247, 7293, 12619, 21805, 37856, 65695, 114401, 199280, 347944, 607959, 1064130, 1864083, 3269948, 5740840, 10090148, 17748870, 31250297, 55063603, 97102485, 171355485, 302605780, 534729160, 945513850

Offset

1,4

Comments

The sequence A001642 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)*A001642(n/d).

Example

u(7) = 9 since a map whose periodic points are counted by A001642 would have 1 fixed point and 64 points of period 7, hence 9 orbits of length 7.

Prog

(PARI) a001642(n) = if(n<0, 0, polcoeff(x*(1+2*x+4*x^3+5*x^4)/(1-x-x^2-x^4-x^5)+x*O(x^n), n));
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001642(n/d)); \\ Michel Marcus, Sep 11 2017

Crossrefs

Cf. A001642, A060164, A060165, A060166, A060168, A060169, A060170, A060171, A060171.

Sequence in context: A068370 A364467 A357488 * A234588 A118550 A126697

Adjacent sequences: A060164 A060165 A060166 * A060168 A060169 A060170

Keyword

easy,nonn

Author

Thomas Ward, Mar 13 2001

Extensions

More terms from Michel Marcus, Sep 11 2017

Status

approved