A060172 Number of orbits of length n under a map whose periodic points are counted by A027306.

1, 1, 1, 2, 3, 6, 9, 19, 28, 62, 93, 205, 315, 703, 1091, 2440, 3855, 8616, 13797, 30801, 49929, 111311, 182361, 405751, 671088, 1490409, 2485504, 5509504, 9256395, 20480421, 34636833, 76499520, 130150493, 286960946, 490853403, 1080476338, 1857283155, 4081876927, 7048151355

Offset

1,4

Comments

The sequence A027306 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..39.

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)*A027306(n/d).

Example

u(7) = 9 since the map whose periodic points are counted by A027306 has 1 fixed point and 64 points of period 7, hence 9 orbits of length 7.

Prog

(PARI) a027306(n) = (2^n + if(n%2, 0, binomial(n, n/2)))/2;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a027306(n/d)); \\ Michel Marcus, Sep 11 2017

Crossrefs

Cf. A027306, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060173.

Sequence in context: A018741 A161704 A011962 * A193196 A319755 A309807

Adjacent sequences: A060169 A060170 A060171 * A060173 A060174 A060175

Keyword

easy,nonn

Author

Thomas Ward, Mar 13 2001

Extensions

More terms from Michel Marcus, Sep 11 2017

Status

approved