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Number of orbits of length n under a map whose periodic points are counted by A027306.
6

%I #8 Feb 19 2021 20:10:00

%S 1,1,1,2,3,6,9,19,28,62,93,205,315,703,1091,2440,3855,8616,13797,

%T 30801,49929,111311,182361,405751,671088,1490409,2485504,5509504,

%U 9256395,20480421,34636833,76499520,130150493,286960946,490853403,1080476338,1857283155,4081876927,7048151355

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

%C 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.

%H Y. Puri and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%H Yash Puri and Thomas Ward, <a href="http://www.fq.math.ca/Scanned/39-5/puri.pdf">A dynamical property unique to the Lucas sequence</a>, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

%F a(n) = (1/n)* Sum_{ d divides n } mu(d)*A027306(n/d).

%e 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.

%o (PARI) a027306(n) = (2^n + if(n%2, 0, binomial(n, n/2)))/2;

%o a(n) = (1/n)*sumdiv(n, d, moebius(d)*a027306(n/d)); \\ _Michel Marcus_, Sep 11 2017

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

%K easy,nonn

%O 1,4

%A _Thomas Ward_, Mar 13 2001

%E More terms from _Michel Marcus_, Sep 11 2017