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A056283 Number of n-bead necklaces with exactly three different colored beads. 7


%S 0,0,2,9,30,91,258,729,2018,5613,15546,43315,120750,338259,950062,

%T 2678499,7573350,21480739,61088874,174184755,497812638,1425847623,

%U 4092087522,11765822365,33887517870,97756387365,282414624746,816999710223,2366509198350,6862930841141

%N Number of n-bead necklaces with exactly three different colored beads.

%C Turning over the necklace is not allowed.

%D 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]

%H Alois P. Heinz, <a href="/A056283/b056283.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A001867(n) - 3*A000031(n) + 3.

%F From _Robert A. Russell_, Sep 26 2018: (Start)

%F a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=3 is the number of colors and S2 is the Stirling subset number A008277.

%F G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=3 is the number of colors. (End)

%e For n=3, the two necklaces are ABC and ACB.

%t k=3; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* _Robert A. Russell_, Sep 26 2018 *)

%Y Cf. A000031, A001867, A052823.

%Y Column k=3 of A087854.

%K nonn

%O 1,3

%A _Marks R. Nester_

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