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

Author

Thomas Ward, Mar 21 2001

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Column 13 of A074650.

Cf. A001022.

Programs

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 *)

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

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