%I #24 Aug 22 2017 20:53:13
%S 3,3,6,12,24,42,78,144,234,456,726,1392,2184,4290,6528,12960,19680,
%T 39078,59046,117600,177060,353562,531438,1061280,1594296,3186456,
%U 4782726,9561552,14348904,28690752,43046718
%N Number of primitive (period n) periodic palindromes using a maximum of three different symbols.
%C For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
%C Number of aperiodic necklaces with three colors that are the same when turned over and hence have reflectional symmetry but no rotational symmetry. - _Herbert Kociemba_, Nov 29 2016
%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]
%F a(n) = Sum_{d|n} mu(d)*A038754(n/d+1).
%F From _Herbert Kociemba_, Nov 29 2016: (Start)
%F More generally, gf(k) is the g.f. for the number of necklaces with reflectional symmetry but no rotational symmetry and beads of k colors.
%F gf(k): Sum_{n>=1} mu(n)*Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)). (End)
%t mx=40;gf[x_,k_]:=Sum[ MoebiusMu[n]*Sum[Binomial[k,i]x^(n i),{i,0,2}]/( 1-k x^(2n)),{n,mx}]; CoefficientList[Series[gf[x,3],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 29 2016 *)
%Y Column 3 of A284856.
%Y Cf. A056459.
%K nonn
%O 1,1
%A _Marks R. Nester_