A060218 Number of orbits of length n under the full 15-shift (whose periodic points are counted by A001024).
15, 105, 1120, 12600, 151872, 1897840, 24408480, 320355000, 4271484000, 57664963104, 786341441760, 10812193870800, 149707312950720, 2085208989609360, 29192926025339776, 410525522071875000, 5795654431511374080, 82105104444274758000, 1166756747396368729440, 16626283650369421872480
Offset: 1
Examples
a(2)=105 since there are 225 points of period 2 in the full 15-shift and 15 fixed points, so there must be (225-15)/2 = 105 orbits of length 2.
Links
- Robert Israel, Table of n, a(n) for n = 1..851
- 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
A060218:= func< n | (&+[MoebiusMu(d)*15^Floor(n/d): d in Divisors(n)])/n >; [A060218(n): n in [1..40]]; // G. C. Greubel, Aug 01 2024
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Maple
f:= n -> 1/n*add(numtheory:-mobius(d)*15^(n/d), d = numtheory:-divisors(n)): map(f, [$1..30]); # Robert Israel, Oct 28 2018
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Mathematica
A060218[n_]:= DivisorSum[n, MoebiusMu[#]*15^(n/#) &]/n; Table[A060218[n], {n, 40}] (* G. C. Greubel, Aug 01 2024 *)
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PARI
a001024(n) = 15^n; a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ Michel Marcus, Sep 11 2017
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SageMath
def A060218(n): return sum(moebius(k)*15^(n//k) for k in (1..n) if (k).divides(n))/n [A060218(n) for n in range(1,41)] # G. C. Greubel, Aug 01 2024
Formula
a(n) = (1/n)* Sum_{d|n} mu(d)*A001024(n/d).
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 15*x^k))/k. - Ilya Gutkovskiy, May 19 2019
Extensions
More terms from Michel Marcus, Sep 11 2017
Comments