A060221 Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).

18, 153, 1938, 26163, 377910, 5667681, 87460002, 1377481950, 22039920504, 357046533675, 5842582734474, 96402612275775, 1601766528128550, 26772383354990049, 449776041098370870, 7589970692848393200, 128583032925805678350, 2185911559727674682148, 37275544492386193492506

Offset: 1

Author

Thomas Ward, Mar 21 2001

Keywords

• easy
• nonn

Comments

Number of Lyndon words (aperiodic necklaces) with n beads of 18 colors. - Andrew Howroyd, Dec 10 2017

Examples

			a(2)=153 since there are 324 points of period 2 in the full 18-shift and 18 fixed points, so there must be (324-18)/2 = 153 orbits of length 2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..792
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
• T. Ward, Exactly realizable sequences

Crossrefs

Column 18 of A074650.

Cf. A001027, A008683.

Programs

Magma

A060221:= func< n | (1/n)*(&+[MoebiusMu(d)*(18)^Floor(n/d): d in Divisors(n)]) >;
[A060221(n): n in [1..40]]; // G. C. Greubel, Sep 13 2024

Mathematica

A060221[n_]:= DivisorSum[n, (18)^(n/#)*MoebiusMu[#] &]/n;
Table[A060221[n], {n, 40}] (* G. C. Greubel, Sep 13 2024 *)

PARI

a001027(n) = 18^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001027(n/d)); \\ Michel Marcus, Sep 11 2017

SageMath

def A060221(n): return (1/n)*sum(moebius(k)*(18)^(n/k) for k in (1..n) if (k).divides(n))
[A060221(n) for n in range(1,41)] # G. C. Greubel, Sep 13 2024

Formula

a(n) = (1/n)* Sum_{d|n} mu(d)*A001027(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 18*x^k))/k. - Ilya Gutkovskiy, May 20 2019

Extensions

More terms from Michel Marcus, Sep 11 2017