login
A060222
Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).
2
19, 171, 2280, 32490, 495216, 7839780, 127695960, 2122929090, 35854187880, 613106378136, 10590023536200, 184442905990860, 3234844881712080, 57071906063500860, 1012075135324821024, 18027588346914850290, 322375697516753069760, 5784852794310472599780, 104127350297911241532840
OFFSET
1,1
COMMENTS
Number of monic irreducible polynomials of degree n over GF(19). - Andrew Howroyd, Dec 10 2017
LINKS
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.
FORMULA
a(n) = (1/n)* Sum_{d|n} mu(d)*A001029(n/d).
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 19*x^k))/k. - Ilya Gutkovskiy, May 20 2019
EXAMPLE
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.
MATHEMATICA
a[n_]:=(1/n) Sum[MoebiusMu[d] 19^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* Vincenzo Librandi, Sep 19 2017 *)
PROG
(PARI) a001029(n) = 19^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001029(n/d)); \\ Michel Marcus, Sep 11 2017
(Magma)
A060222:= func< n | (1/n)*(&+[MoebiusMu(d)*(19)^Floor(n/d): d in Divisors(n)]) >;
[A060222(n): n in [1..40]]; // G. C. Greubel, Sep 23 2024
(SageMath)
def A060222(n): return (1/n)*sum(moebius(k)*(19)^(n/k) for k in (1..n) if (k).divides(n))
[A060222(n) for n in range(1, 41)] # G. C. Greubel, Sep 23 2024
CROSSREFS
Column 19 of A074650.
Cf. A001029.
Sequence in context: A282288 A022614 A322878 * A041690 A217698 A172642
KEYWORD
easy,nonn
AUTHOR
Thomas Ward, Mar 21 2001
EXTENSIONS
More terms from Michel Marcus, Sep 11 2017
STATUS
approved