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A056288 Number of primitive (period n) n-bead necklaces with exactly three different colored beads.

%I A056288 #17 Jun 06 2018 14:33:52
%S A056288 0,0,2,9,30,89,258,720,2016,5583,15546,43215,120750,338001,950030,
%T A056288 2677770,7573350,21478632,61088874,174179133,497812378,1425832077,
%U A056288 4092087522,11765778330,33887517840,97756266615,282414622728,816999371955,2366509198350,6862929885407
%N A056288 Number of primitive (period n) n-bead necklaces with exactly three different colored beads.
%C A056288 Turning over the necklace is not allowed.
%D A056288 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 A056288 Alois P. Heinz, <a href="/A056288/b056288.txt">Table of n, a(n) for n = 1..1000</a>
%F A056288 Sum mu(d)*A056283(n/d) where d|n.
%p A056288 with(numtheory):
%p A056288 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A056288       add(mobius(n/d)*k^d, d=divisors(n))/n)
%p A056288     end:
%p A056288 a:= n-> add(b(n, 3-j)*binomial(3, j)*(-1)^j, j=0..3):
%p A056288 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jan 25 2015
%t A056288 b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^#&]/n];
%t A056288 a[n_] := Sum[b[n, 3 - j]*Binomial[3, j]*(-1)^j, {j, 0, 3}];
%t A056288 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Jun 06 2018, after _Alois P. Heinz_ *)
%Y A056288 Cf. A027376.
%Y A056288 Column k=3 of A254040.
%K A056288 nonn
%O A056288 1,3
%A A056288 _Marks R. Nester_