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A060217 Number of orbits of length n under the full 14-shift (whose periodic points are counted by A001023).

%I A060217 #27 Jan 05 2025 19:51:36
%S A060217 14,91,910,9555,107562,1254435,15059070,184468830,2295671560,
%T A060217 28925411697,368142288150,4724492067295,61054982558010,
%U A060217 793714765724595,10371206370484778,136122083520848880,1793608631137129170,23715491899442676060,314542313628890231430,4183412771249777343369
%N A060217 Number of orbits of length n under the full 14-shift (whose periodic points are counted by A001023).
%C A060217 Number of Lyndon words (aperiodic necklaces) with n beads of 14 colors. - _Andrew Howroyd_, Dec 10 2017
%H A060217 G. C. Greubel, <a href="/A060217/b060217.txt">Table of n, a(n) for n = 1..870</a>
%H A060217 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 A060217 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 A060217 T. Ward, <a href="http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>
%F A060217 a(n) = (1/n)* Sum_{d|n} mu(d)*A001023(n/d).
%F A060217 G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 14*x^k))/k. - _Ilya Gutkovskiy_, May 19 2019
%e A060217 a(2)=91 since there are 196 points of period 2 in the full 14-shift and 14 fixed points, so there must be (196-14)/2 = 91 orbits of length 2.
%t A060217 A060217[n_]:= DivisorSum[n, MoebiusMu[#]*14^(n/#) &]/n;
%t A060217 Table[A060217[n], {n,40}] (* _G. C. Greubel_, Aug 01 2024 *)
%o A060217 (PARI) a001023(n) = 14^n;
%o A060217 a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001023(n/d)); \\ _Michel Marcus_, Sep 11 2017
%o A060217 (Magma)
%o A060217 A060217:= func< n | (&+[MoebiusMu(d)*14^Floor(n/d): d in Divisors(n)])/n >;
%o A060217 [A060217(n): n in [1..40]]; // _G. C. Greubel_, Aug 01 2024
%o A060217 (SageMath)
%o A060217 def A060217(n): return sum(moebius(k)*14^(n//k) for k in (1..n) if (k).divides(n))/n
%o A060217 [A060217(n) for n in range(1,41)] # _G. C. Greubel_, Aug 01 2024
%Y A060217 Column 14 of A074650.
%Y A060217 Cf. A001023.
%K A060217 easy,nonn
%O A060217 1,1
%A A060217 _Thomas Ward_, Mar 21 2001
%E A060217 More terms from _Michel Marcus_, Sep 11 2017