A056494 Number of primitive (period n) periodic palindromes using a maximum of three different symbols.

3, 3, 6, 12, 24, 42, 78, 144, 234, 456, 726, 1392, 2184, 4290, 6528, 12960, 19680, 39078, 59046, 117600, 177060, 353562, 531438, 1061280, 1594296, 3186456, 4782726, 9561552, 14348904, 28690752, 43046718

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

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

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]

Crossrefs

Column 3 of A284856.

Cf. A056459.

Programs

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,3],{x,0,mx}],x] (* Herbert Kociemba, Nov 29 2016 *)

Formula

a(n) = Sum_{d|n} mu(d)*A038754(n/d+1).

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)