A060219 Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025).

16, 120, 1360, 16320, 209712, 2795480, 38347920, 536862720, 7635496960, 109951057896, 1599289640400, 23456246655680, 346430740566960, 5146970983535160, 76861433640386288, 1152921504338411520, 17361641481138401520, 262353693488939386880, 3976729669784964390480

Offset: 1

Author

Thomas Ward, Mar 21 2001

Keywords

• easy
• nonn

Comments

Number of monic irreducible polynomials of degree n over GF(16). - Robert Israel, Jan 07 2015

Number of Lyndon words (aperiodic necklaces) with n beads of 16 colors. - Andrew Howroyd, Dec 10 2017

Examples

			a(2)=120 since there are 256 points of period 2 in the full 16-shift and 16 fixed points, so there must be (256-16)/2 = 120 orbits of length 2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..825
• 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 16 of A074650.

Cf. A001025.

Programs

Magma

A060219:= func< n | (&+[MoebiusMu(d)*16^Floor(n/d): d in Divisors(n)])/n >;
[A060219(n): n in [1..40]]; // G. C. Greubel, Aug 01 2024

Maple

f:= (n,p) -> add(numtheory:-mobius(d)*p^(n/d),d=numtheory:-divisors(n))/n:
seq(f(n,16),n=1..30); # Robert Israel, Jan 07 2015

Mathematica

A060219[n_]:= DivisorSum[n, MoebiusMu[#]*16^(n/#) &]/n;Table[A060219[n], {n, 40}] (* G. C. Greubel, Aug 01 2024 *)

PARI

a(n) = sumdiv(n, d, moebius(d)*16^(n/d))/n; \\ Michel Marcus, Jan 07 2015

SageMath

def A060219(n): return sum(moebius(k)*16^(n//k) for k in (1..n) if (k).divides(n))/n
[A060219(n) for n in range(1, 41)] # G. C. Greubel, Aug 01 2024

Formula

a(n) = (1/n)* Sum_{d|n} mu(d)*16^(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 16*x^k))/k. - Ilya Gutkovskiy, May 19 2019

Extensions

Terms a(17) onward added by G. C. Greubel, Aug 01 2024