A060222 Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).
19, 171, 2280, 32490, 495216, 7839780, 127695960, 2122929090, 35854187880, 613106378136, 10590023536200, 184442905990860, 3234844881712080, 57071906063500860, 1012075135324821024, 18027588346914850290, 322375697516753069760, 5784852794310472599780, 104127350297911241532840
Offset: 1
Examples
a(2)=171 since there are 361 points of period 2 in the full 19-shift and 19 fixed points, so there must be (361-19)/2 = 171 orbits of length 2.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..799
- 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.
- T. Ward, Exactly realizable sequences
Programs
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Magma
A060222:= func< n | (1/n)*(&+[MoebiusMu(d)*(19)^Floor(n/d): d in Divisors(n)]) >; [A060222(n): n in [1..40]]; // G. C. Greubel, Sep 23 2024
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Mathematica
a[n_]:=(1/n) Sum[MoebiusMu[d] 19^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* Vincenzo Librandi, Sep 19 2017 *) -
PARI
a001029(n) = 19^n; a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001029(n/d)); \\ Michel Marcus, Sep 11 2017
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SageMath
def A060222(n): return (1/n)*sum(moebius(k)*(19)^(n/k) for k in (1..n) if (k).divides(n)) [A060222(n) for n in range(1, 41)] # G. C. Greubel, Sep 23 2024
Formula
a(n) = (1/n)* Sum_{d|n} mu(d)*A001029(n/d).
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 19*x^k))/k. - Ilya Gutkovskiy, May 20 2019
Extensions
More terms from Michel Marcus, Sep 11 2017
Comments