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A032191
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Number of necklaces with 6 black beads and n-6 white beads.
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11
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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
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OFFSET
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6,3
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COMMENTS
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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 as this sequence's name: 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 disregard the labels. - Wolfdieter Lang, Feb 15 2005
The g.f. of the CIK[k] transform of the sequence (b(n): n>=1), which has g.f. B(x) = Sum_{n>=1} b(n)*x^n, is CIK[k](x) = (1/k)*Sum_{d|k} phi(d)*B(x^d)^{k/d}. Here, k = 6, b(n) = 1 for all n >= 1, and B(x) = x/(1-x), from which we get another proof of the g.f.s given below. - Petros Hadjicostas, Jan 07 2018
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,4,-3,-3,4,1,-1,-3,1,2,-1).
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FORMULA
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"CIK[ 6 ]" (necklace, indistinct, unlabeled, 6 parts) transform of 1, 1, 1, 1, ...
G.f.: (1-x+x^2+4*x^3+2*x^4+3*x^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; see Comments entry above). - Wolfdieter Lang, Feb 15 2005
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
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EXAMPLE
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We explain why a(8) = 4. According to the theory of transforms by C. G. Bower, given in the weblink above, a(8) is the number of ways of arranging 6 indistinct unlabeled boxes (that may differ only in their size) as a necklace, on a circle, such that the total number of balls in all of them is 8. There are 4 ways for doing that on a circle: 311111, 221111, 212111, and 211211.
To translate these configurations of boxes into necklaces with 8 beads, 6 of them black and 2 of them white, we modify an idea given above by W. Lang. We replace each box that has m balls with a black bead followed by m-1 white beads. The four examples above become BWWBBBBB, BWBWBBBB, BWBBWBBB, and BWBBBWBB.
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MATHEMATICA
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k = 6; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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