A001869 Number of n-bead necklaces with 5 colors. (Formerly M3860 N1582)

1, 5, 15, 45, 165, 629, 2635, 11165, 48915, 217045, 976887, 4438925, 20346485, 93900245, 435970995, 2034505661, 9536767665, 44878791365, 211927736135, 1003867701485, 4768372070757, 22706531350485, 108372083629275, 518301258916445

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 5^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

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 5

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

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

Eric Weisstein's World of Mathematics, Necklace.

Index entries for sequences related to necklaces

Formula

a(n) = (1/n)*Sum_{d|n} phi(d)*5^(n/d), n > 0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 5*x^n)/n. - Herbert Kociemba, Nov 01 2016

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 5^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021

a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 5^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

Mathematica

CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-5*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 01 2016 *)
k=5; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* Robert A. Russell, Sep 21 2018 *)

Prog

(PARI) a(n) = if (n, sumdiv(n, d, eulerphi(d)*5^(n/d))/n, 1); \\ Michel Marcus, Nov 01 2016

Crossrefs

Column 5 of A075195.

Cf. A054612.

Sequence in context: A207096 A035069 A176611 * A058425 A079798 A344814

Adjacent sequences: A001866 A001867 A001868 * A001870 A001871 A001872

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved