A001867 Number of n-bead necklaces with 3 colors. (Formerly M2548 N1008)

1, 3, 6, 11, 24, 51, 130, 315, 834, 2195, 5934, 16107, 44368, 122643, 341802, 956635, 2690844, 7596483, 21524542, 61171659, 174342216, 498112275, 1426419858, 4093181691, 11767920118, 33891544419, 97764131646, 282429537947, 817028472960, 2366564736723

Offset

0,2

Comments

From Richard L. Ollerton, May 07 2021: (Start)

Here, as in A000031, turning over is not allowed.

(1/n) * Dirichlet convolution of phi(n) and 3^n, n>0. (End)

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 162.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).

Links

T. D. Noe, Table of n, a(n) for n = 0..200

Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See pp. 6, 20.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

V. E. Hoggatt, The Fifth Oldie from the Vault. Problem B-415, Elementary Problems and Solutions, Fibonacci Quart. 59(3) (2021), 274-275.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 3.

Juhani Karhumäki, Svetlana Puzynina, Michaël Rao, and Markus A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.

Shisheng Li, SB(3,n) has no Hamiltonian cycle when n is even: a sign-of-permutation proof, with extension to all odd m == 3 (mod 4), arXiv:2605.09489 [math.CO], 2026. See pp. 5, 7.

J. Riordan, Letter to N. J. A. Sloane, Jul. 1978.

Index entries for sequences related to necklaces

Formula

a(n) = (1/n)*Sum_{d|n} phi(d)*3^(n/d), n>0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 3*x^n)/n. - Herbert Kociemba, Nov 01 2016

a(n) ~ 3^n/n. - Vaclav Kotesovec, Nov 01 2016

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 3^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 3^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

Maple

with(numtheory): A001867:= n-> `if` (n=0, 1, add (phi(d)* 3^(n/d), d=divisors(n))/n): seq (A001867(n), n=0..40);
spec := [N, {N=Cycle(bead), bead=Union(R, G, B), R=Atom, B=Atom, G=Atom}]; [seq(combstruct[count](spec, size=n), n=1..40)];

Mathematica

Prepend[Table[CyclicGroupIndex[n, t]/.Table[t[i]->3, {i, 1, n}], {n, 1, 28}], 1]  (* Geoffrey Critzer, Sep 16 2011 *)
(* Alternative: *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-3*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 01 2016 *)
(* Alternative: *)
k=3; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* Robert A. Russell, Sep 21 2018 *)

Prog

(PARI) a(n)=if (n==0, 1, 1/n * sumdiv(n, d, eulerphi(d)*3^(n/d) )); /* Joerg Arndt, Jul 04 2011 */

Crossrefs

Column 3 of A075195.

Cf. A054610.

Sequence in context: A390564 A346050 A319636 * A369691 A000998 A331536

Adjacent sequences: A001864 A001865 A001866 * A001868 A001869 A001870

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved