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A060216 Number of orbits of length n under the full 13-shift (whose periodic points are counted by A001022). 2
13, 78, 728, 7098, 74256, 804076, 8964072, 101962770, 1178277464, 13785812040, 162923672184, 1941506688940, 23298085122480, 281241165925044, 3412392867581152, 41588538022965570 (list; graph; refs; listen; history; text; internal format)
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 [(1/n) * sum([mobius(d) * 13**(n/d) for d in divisors(n)]) for n in xrange(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 (t.ward(AT)uea.ac.uk), Mar 21 2001

STATUS

approved

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Last modified October 19 15:50 EDT 2019. Contains 328223 sequences. (Running on oeis4.)