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A060221 Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).

%I A060221 #23 Jan 05 2025 19:51:36
%S A060221 18,153,1938,26163,377910,5667681,87460002,1377481950,22039920504,
%T A060221 357046533675,5842582734474,96402612275775,1601766528128550,
%U A060221 26772383354990049,449776041098370870,7589970692848393200,128583032925805678350,2185911559727674682148,37275544492386193492506
%N A060221 Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).
%C A060221 Number of Lyndon words (aperiodic necklaces) with n beads of 18 colors. - _Andrew Howroyd_, Dec 10 2017
%H A060221 G. C. Greubel, <a href="/A060221/b060221.txt">Table of n, a(n) for n = 1..792</a>
%H A060221 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 A060221 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 A060221 T. Ward, <a href="http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>
%F A060221 a(n) = (1/n)* Sum_{d|n} mu(d)*A001027(n/d).
%F A060221 G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 18*x^k))/k. - _Ilya Gutkovskiy_, May 20 2019
%e A060221 a(2)=153 since there are 324 points of period 2 in the full 18-shift and 18 fixed points, so there must be (324-18)/2 = 153 orbits of length 2.
%t A060221 A060221[n_]:= DivisorSum[n, (18)^(n/#)*MoebiusMu[#] &]/n;
%t A060221 Table[A060221[n], {n, 40}] (* _G. C. Greubel_, Sep 13 2024 *)
%o A060221 (PARI) a001027(n) = 18^n;
%o A060221 a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001027(n/d)); \\ _Michel Marcus_, Sep 11 2017
%o A060221 (Magma)
%o A060221 A060221:= func< n | (1/n)*(&+[MoebiusMu(d)*(18)^Floor(n/d): d in Divisors(n)]) >;
%o A060221 [A060221(n): n in [1..40]]; // _G. C. Greubel_, Sep 13 2024
%o A060221 (SageMath)
%o A060221 def A060221(n): return (1/n)*sum(moebius(k)*(18)^(n/k) for k in (1..n) if (k).divides(n))
%o A060221 [A060221(n) for n in range(1,41)] # _G. C. Greubel_, Sep 13 2024
%Y A060221 Column 18 of A074650.
%Y A060221 Cf. A001027, A008683.
%K A060221 easy,nonn
%O A060221 1,1
%A A060221 _Thomas Ward_, Mar 21 2001
%E A060221 More terms from _Michel Marcus_, Sep 11 2017