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A056283
Number of n-bead necklaces with exactly three different colored beads.
7
0, 0, 2, 9, 30, 91, 258, 729, 2018, 5613, 15546, 43315, 120750, 338259, 950062, 2678499, 7573350, 21480739, 61088874, 174184755, 497812638, 1425847623, 4092087522, 11765822365, 33887517870, 97756387365, 282414624746, 816999710223, 2366509198350, 6862930841141
OFFSET
1,3
COMMENTS
Turning over the necklace is not allowed.
REFERENCES
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]
LINKS
FORMULA
a(n) = A001867(n) - 3*A000031(n) + 3.
From Robert A. Russell, Sep 26 2018: (Start)
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.
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)
EXAMPLE
For n=3, the two necklaces are ABC and ACB.
MATHEMATICA
k=3; Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}] (* Robert A. Russell, Sep 26 2018 *)
CROSSREFS
Column k=3 of A087854.
Sequence in context: A056288 A261174 A273652 * A192518 A277241 A201164
KEYWORD
nonn
STATUS
approved