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A060222 Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).

%I A060222 #28 Jan 05 2025 19:51:36
%S A060222 19,171,2280,32490,495216,7839780,127695960,2122929090,35854187880,
%T A060222 613106378136,10590023536200,184442905990860,3234844881712080,
%U A060222 57071906063500860,1012075135324821024,18027588346914850290,322375697516753069760,5784852794310472599780,104127350297911241532840
%N A060222 Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).
%C A060222 Number of monic irreducible polynomials of degree n over GF(19). - _Andrew Howroyd_, Dec 10 2017
%H A060222 Vincenzo Librandi, <a href="/A060222/b060222.txt">Table of n, a(n) for n = 1..799</a>
%H A060222 Y. Puri and T. Ward, <a href="http://www.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 A060222 Yash Puri and Thomas Ward, <a href="https://web.archive.org/web/2024*/https://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.
%H A060222 T. Ward, <a href="https://web.archive.org/web/20110921045528/http://www.uea.ac.uk:80/~h720/research/files/integersequences.html">Exactly realizable sequences</a>
%F A060222 a(n) = (1/n)* Sum_{d|n} mu(d)*A001029(n/d).
%F A060222 G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 19*x^k))/k. - _Ilya Gutkovskiy_, May 20 2019
%e A060222 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.
%t A060222 a[n_]:=(1/n) Sum[MoebiusMu[d] 19^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* _Vincenzo Librandi_, Sep 19 2017 *)
%o A060222 (PARI) a001029(n) = 19^n;
%o A060222 a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001029(n/d)); \\ _Michel Marcus_, Sep 11 2017
%o A060222 (Magma)
%o A060222 A060222:= func< n | (1/n)*(&+[MoebiusMu(d)*(19)^Floor(n/d): d in Divisors(n)]) >;
%o A060222 [A060222(n): n in [1..40]]; // _G. C. Greubel_, Sep 23 2024
%o A060222 (SageMath)
%o A060222 def A060222(n): return (1/n)*sum(moebius(k)*(19)^(n/k) for k in (1..n) if (k).divides(n))
%o A060222 [A060222(n) for n in range(1, 41)] # _G. C. Greubel_, Sep 23 2024
%Y A060222 Column 19 of A074650.
%Y A060222 Cf. A001029.
%K A060222 easy,nonn
%O A060222 1,1
%A A060222 _Thomas Ward_, Mar 21 2001
%E A060222 More terms from _Michel Marcus_, Sep 11 2017