OFFSET
1,1
COMMENTS
Number of monic irreducible polynomials of degree n over GF(13). - Robert Israel, Jan 07 2015
Number of Lyndon words (aperiodic necklaces) with n beads of 13 colors. - Andrew Howroyd, Dec 10 2017
LINKS
Indranil Ghosh, Table of n, a(n) for n = 1..100
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Exactly realizable sequences
FORMULA
a(n) = (1/n)* Sum_{d|n} mu(d) 13^(n/d).
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 13*x^k))/k. - Ilya Gutkovskiy, May 19 2019
EXAMPLE
a(2)=78 since there are 169 points of period 2 in the full 13-shift and 13 fixed points, so there must be (169-13)/2 = 78 orbits of length 2.
MAPLE
f:= n -> add(numtheory:-mobius(d)*13^(n/d), d=numtheory:-divisors(n))/n;
seq(f(n), n=1..100); # Robert Israel, Jan 07 2015
MATHEMATICA
a[n_]:=(1/n) * Sum[MoebiusMu[d] *13^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* Indranil Ghosh, Mar 26 2017 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(d)*13^(n/d))/n; \\ Michel Marcus, Jan 07 2015
(Python)
from sympy import divisors, mobius
print([sum(mobius(d) * 13**(n//d) for d in divisors(n))//n for n in range(1, 21)]) # Indranil Ghosh, Mar 26 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Ward, Mar 21 2001
STATUS
approved