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A054625
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Number of n-bead necklaces with 6 colors.
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8
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1, 6, 21, 76, 336, 1560, 7826, 39996, 210126, 1119796, 6047412, 32981556, 181402676, 1004668776, 5597460306, 31345666736, 176319474366, 995685849696, 5642220380006, 32071565263716, 182807925027504, 1044616697187576, 5982804736593846
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OFFSET
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0,2
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LINKS
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Eric Weisstein's World of Mathematics, Necklace.
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FORMULA
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a(n) = (1/n)*Sum_{d|n} phi(d)*6^(n/d), n > 0.
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 6^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021
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EXAMPLE
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G.f. = 1 + 6*x + 21*x^2 + 76*x^3 + 336*x^4 + 1650*x^5 + 7826*x^6 + 39996*x^7 + ...
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MAPLE
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with(combstruct):A:=[N, {N=Cycle(Union(Z$6))}, unlabeled]: seq(count(A, size=n), n=0..22); # Zerinvary Lajos, Dec 05 2007
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MATHEMATICA
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f[n_] := Block[{d = Divisors@ n}, Total[EulerPhi[d]*6^(n/d)]/n]; f[0] = 1; Array[f, 23, 0] (* Robert G. Wilson v, Jan 01 2013 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-6*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 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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EXTENSIONS
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STATUS
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approved
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