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

1,1

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]

Links

Table of n, a(n) for n=1..31.

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)

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 *)

Crossrefs

Column 3 of A284856.

Cf. A056459.

Sequence in context: A050067 A309399 A046875 * A168076 A168073 A231829

Adjacent sequences: A056491 A056492 A056493 * A056495 A056496 A056497

Keyword

nonn

Author

Marks R. Nester

Status

approved