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%I #16 Aug 22 2017 20:53:13
%S 4,6,12,30,60,138,252,600,1008,2490,4092,10050,16380,40698,65460,
%T 163200,262140,654192,1048572,2618850,4194036,10481658,16777212,
%U 41932200,67108800,167755770,268434432,671047650
%N Number of primitive (period n) periodic palindromes using a maximum of four different symbols.
%C Number of aperiodic necklaces with four 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)*A056486(n/d).
%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)
%e For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
%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,4],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 29 2016 *)
%Y Column 4 of A284856.
%Y Cf. A056460.
%K nonn
%O 1,1
%A _Marks R. Nester_
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