A060170 Number of orbits of length n under the map whose periodic points are counted by A005809.

3, 6, 27, 120, 600, 3078, 16611, 91872, 520749, 3004200, 17594247, 104304888, 624801957, 3775722342, 22991161500, 140928011136, 868886416866, 5384796881850, 33525472069563, 209592223788000, 1315211209630794, 8281053081282894, 52301607644921259, 331260902534858976, 2103541885645955625, 13389670112374830378

Offset

1,1

Comments

The sequence A005809 records the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

a(n) is divisible by n (cf. A268617), 2*a(n) is divisible by n^2 (cf. A268618).

Links

Table of n, a(n) for n=1..26.

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} A008683(n/d)*A005809(d).

Example

a(3) = 27 since a map whose periodic points are counted by A005809 has 3 fixed points and 84 points of period 3, hence 27 orbits of length 3.

Prog

(PARI) a(n) = sumdiv(n, d, moebius(n/d)*binomial(3*d, d))/n; \\ Michel Marcus, Sep 10 2017

Crossrefs

Cf. A005809, A060164, A060165, A060166, A060167, A060168, A060179, A060171, A060171, A060172, A060173.

Sequence in context: A215400 A215394 A222625 * A223143 A372039 A270889

Adjacent sequences: A060167 A060168 A060169 * A060171 A060172 A060173

Keyword

easy,nonn

Author

Thomas Ward, Mar 13 2001

Extensions

Edited by Max Alekseyev, Feb 09 2016

Status

approved