A060173 Number of orbits of length n under a map whose periodic points are counted by A056045.

1, 1, 1, 2, 1, 6, 1, 12, 10, 30, 1, 139, 1, 252, 231, 920, 1, 3780, 1, 10250, 5601, 32076, 1, 149390, 2126, 400036, 173692, 1475642, 1, 6196651, 1, 19113136, 5864915, 68635494, 201405, 289525026, 1, 930138540, 208267554, 3469290971, 1, 14075005210, 1, 47994721225, 7683440470

Offset

1,4

Comments

The sequence A056045 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.

Links

Table of n, a(n) for n=1..45.

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)*A056045(n/d).

Example

a(7) = 1 since the map whose periodic points are counted by A056045 has 1 fixed point and 8 points of period 7, hence 1 orbits of length 7.

Prog

(PARI) a056045(n) = sumdiv(n, d, binomial(n, d));
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a056045(n/d)); \\ Michel Marcus, Sep 11 2017

Crossrefs

Cf. A056045, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060172.

Sequence in context: A053785 A233809 A207536 * A059344 A109193 A322672

Adjacent sequences: A060170 A060171 A060172 * A060174 A060175 A060176

Keyword

easy,nonn

Author

Thomas Ward, Mar 13 2001

Extensions

More terms from Michel Marcus, Sep 11 2017

Status

approved