A060216 Number of orbits of length n under the full 13-shift (whose periodic points are counted by A001022).

13, 78, 728, 7098, 74256, 804076, 8964072, 101962770, 1178277464, 13785812040, 162923672184, 1941506688940, 23298085122480, 281241165925044, 3412392867581152, 41588538022965570

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

Column 13 of A074650.

Cf. A001022.

Sequence in context: A047638 A010929 A022608 * A041318 A142056 A173831

Adjacent sequences: A060213 A060214 A060215 * A060217 A060218 A060219

Keyword

nonn

Author

Thomas Ward, Mar 21 2001

Status

approved