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A056347
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Number of primitive (period n) bracelets using a maximum of six different colored beads.
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1
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6, 15, 50, 210, 882, 4220, 20640, 107100, 563730, 3036411, 16514100, 90778485, 502474350, 2799199380, 15673672238, 88162569180, 497847963690, 2821127257950, 16035812864940, 91404065292036
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OFFSET
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1,1
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COMMENTS
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Turning over will not create a new bracelet.
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REFERENCES
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M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
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LINKS
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FORMULA
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More generally, gf(k) is the g.f. for the number of bracelets with primitive period n and beads of k colors.
gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n + Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End)
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MATHEMATICA
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mx=40; gf[x_, k_]:=Sum[ MoebiusMu[n]*(-Log[1-k*x^n]/n+Sum[Binomial[k, i]x^(n i), {i, 0, 2}]/( 1-k x^(2n)))/2, {n, mx}]; CoefficientList[Series[gf[x, 6], {x, 0, mx}], x] (* Herbert Kociemba, Nov 28 2016 *)
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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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