A060477 Number of orbits of length n in map whose periodic points are A000051.
3, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080, 7710, 14532, 27594, 52377, 99858, 190557, 364722, 698870, 1342176, 2580795, 4971008, 9586395, 18512790, 35790267, 69273666, 134215680, 260300986, 505286415, 981706806, 1908866960, 3714566310
Offset: 1
Examples
a(3)=2 since the 3rd term of A000051 is 9 and the first term is 3.
Links
- 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.
Programs
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PARI
a000051(n) = 2^n+1; a(n) = (1/n)*sumdiv(n, d, moebius(d)*a000051(n/d)); \\ Michel Marcus, Sep 11 2017
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Python
from sympy import mobius, divisors 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
Formula
a(n) = (1/n)* Sum_{d|n} mu(d)*A000051(n/d).