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A032191 Number of necklaces with 6 black beads and n-6 white beads. 8
1, 1, 4, 10, 22, 42, 80, 132, 217, 335, 504, 728, 1038, 1428, 1944, 2586, 3399, 4389, 5620, 7084, 8866, 10966, 13468, 16380, 19811, 23751, 28336, 33566, 39576, 46376, 54132, 62832, 72675, 83661, 95988, 109668, 124936, 141778 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,3

COMMENTS

The g.f. is Z(C_6,x)/x^6, the 6-variate cycle index polynomial for the cyclic group C_6, with substitution x[i]->1/(1-x^i), i=1,...,6. Therefore by Polya enumeration a(n+6) is the number of cyclically inequivalent 6-necklaces whose 6 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_6,x). Note the equivalence of this formulation with the one given in the `Name' line: start with a black 6-necklace (all 6 beads have labels 0). Insert after each of the 6 black beads k white ones if the label was k and then forget about the labels. Wolfdieter Lang, Feb 15 2005.

LINKS

Table of n, a(n) for n=6..43.

C. G. Bower, Transforms (2)

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

Index entries for sequences related to necklaces

FORMULA

"CIK[ 6 ]" (necklace, indistinct, unlabeled, 6 parts) transform of 1, 1, 1, 1...

G.f.: [1-x+x^2+4x^3+2x^4+3x^6+x^7+x^8]/[(1-x)^6(1+x)^3(1+x+x^2)^2(1-x+x^2)] (conjectured). - Ralf Stephan, May 05 2004

G.f.:(x^6)*(1-x+x^2+4*x^3+2*x^4+3*x^6+x^7+x^8)/((1-x)^2*(1-x^2)^2*(1-x^3)*(1-x^6)) (proving the R. Stephan conjecture (with the correct offset) in a different version. W. Lang see above.)

G.f.: (1/6)*x^6*((1-x)^(-6)+(1-x^2)^(-3)+2*(1-x^3)^(-2)+2*(1-x^6)^(-1)). - Herbert Kociemba, Oct 22 2016

MATHEMATICA

k = 6; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

CROSSREFS

Cf. A004526, A007997, A008610, A008646.

Sequence in context: A023609 A055364 A284870 * A065568 A007825 A008256

Adjacent sequences:  A032188 A032189 A032190 * A032192 A032193 A032194

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified August 24 04:54 EDT 2017. Contains 291052 sequences.