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A001869 Number of n-bead necklaces with 5 colors.
(Formerly M3860 N1582)
8
1, 5, 15, 45, 165, 629, 2635, 11165, 48915, 217045, 976887, 4438925, 20346485, 93900245, 435970995, 2034505661, 9536767665, 44878791365, 211927736135, 1003867701485, 4768372070757, 22706531350485, 108372083629275, 518301258916445 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

Juhani Karhumäki, S Puzynina, M Rao, MA Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319, 2016

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

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

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 *)

PROG

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

CROSSREFS

Sequence in context: A207096 A035069 A176611 * A058425 A079798 A037504

Adjacent sequences:  A001866 A001867 A001868 * A001870 A001871 A001872

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 26 04:32 EDT 2017. Contains 289798 sequences.