OFFSET
1,1
COMMENTS
Number of Lyndon words (aperiodic necklaces) with n beads of 15 colors. - Andrew Howroyd, Dec 10 2017
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
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
EXAMPLE
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.
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
MATHEMATICA
A060218[n_]:= DivisorSum[n, MoebiusMu[#]*15^(n/#) &]/n;
Table[A060218[n], {n, 40}] (* G. C. Greubel, Aug 01 2024 *)
PROG
(PARI) a001024(n) = 15^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ Michel Marcus, Sep 11 2017
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Thomas Ward, Mar 21 2001
EXTENSIONS
More terms from Michel Marcus, Sep 11 2017
STATUS
approved