OFFSET
1,1
COMMENTS
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
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]
FORMULA
a(n) = Sum_{d|n} mu(d)*A056486(n/d).
From Herbert Kociemba, Nov 29 2016: (Start)
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.
gf(k): Sum_{n>=1} mu(n)*Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)). (End)
EXAMPLE
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved