%I #16 Aug 22 2017 20:53:13

%S 5,10,20,60,120,340,620,1800,3100,9240,15620,46440,78120,233740,

%T 390480,1170000,1953120,5855900,9765620,29287440,48827480,146468740,

%U 244140620,732373200,1220703000,3662031240

%N Number of primitive (period n) periodic palindromes using a maximum of five different symbols.

%C Number of aperiodic necklaces with five 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)*A056487(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,5],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 29 2016 *)

%Y Column 5 of A284856.

%Y Cf. A056461.

%K nonn

%O 1,1

%A _Marks R. Nester_