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A001868 Number of n-bead necklaces with 4 colors.
(Formerly M3390 N1370)
18
1, 4, 10, 24, 70, 208, 700, 2344, 8230, 29144, 104968, 381304, 1398500, 5162224, 19175140, 71582944, 268439590, 1010580544, 3817763740, 14467258264, 54975633976, 209430787824, 799645010860, 3059510616424, 11728124734500, 45035996273872, 173215372864600, 667199944815064 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 4

Juhani Karhumäki, S. Puzynina, M. Rao, 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

Index entries for sequences related to necklaces

FORMULA

a(n) = (1/n)*Sum_{d|n} phi(d)*4^(n/d), n>0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 4*x^n)/n. - Herbert Kociemba, Nov 01 2016

MAPLE

A001868 := proc(n) local d, s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*4^(n/d); od; RETURN(s/n); fi; end;

MATHEMATICA

a[n_] := (1/n)*Total[ EulerPhi[#]*4^(n/#) &  /@ Divisors[n]]; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Oct 21 2011 *)

mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-4*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 01 2016 *)

k=4; 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)*4^(n/d))/n, 1); \\ Michel Marcus, Nov 01 2016

CROSSREFS

Column 4 of A075195.

Cf. A054611.

Sequence in context: A190169 A212330 A291412 * A217696 A223014 A038783

Adjacent sequences:  A001865 A001866 A001867 * A001869 A001870 A001871

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 23 03:21 EDT 2018. Contains 316519 sequences. (Running on oeis4.)