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Number of orbits of length n in map whose periodic points are A000051.
4

%I #20 Feb 03 2022 13:52:39

%S 3,1,2,3,6,9,18,30,56,99,186,335,630,1161,2182,4080,7710,14532,27594,

%T 52377,99858,190557,364722,698870,1342176,2580795,4971008,9586395,

%U 18512790,35790267,69273666,134215680,260300986,505286415,981706806,1908866960,3714566310

%N Number of orbits of length n in map whose periodic points are A000051.

%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|n} mu(d)*A000051(n/d).

%e a(3)=2 since the 3rd term of A000051 is 9 and the first term is 3.

%o (PARI) a000051(n) = 2^n+1;

%o a(n) = (1/n)*sumdiv(n, d, moebius(d)*a000051(n/d)); \\ _Michel Marcus_, Sep 11 2017

%o (Python)

%o from sympy import mobius, divisors

%o def A060477(n): return sum(mobius(n//d)*(2**d+1) for d in divisors(n,generator=True))//n # _Chai Wah Wu_, Feb 03 2022

%Y Cf. A000051.

%Y Cf. A001037, A059966 (both nearly identical to this sequence).

%Y Cf. A093210.

%K easy,nonn

%O 1,1

%A _Thomas Ward_

%E A048578 replaced by A000051 in name and formula by _Michel Marcus_, Sep 11 2017