A056497 Number of primitive (period n) periodic palindromes using a maximum of six different symbols.

6, 15, 30, 105, 210, 705, 1290, 4410, 7740, 26985, 46650, 162435, 279930, 978465, 1679370, 5874120, 10077690, 35263440, 60466170, 211604295, 362795730, 1269743025, 2176782330, 7618570470, 13060693800

Offset

1,1

Comments

Number of aperiodic necklaces with six 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..25.

Formula

a(n) = Sum_{d|n} mu(d)*A056488(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, 6], {x, 0, mx}], x] (* Herbert Kociemba, Nov 29 2016 *)

Crossrefs

Column 6 of A284856.

Cf. A056462.

Sequence in context: A353695 A232169 A231814 * A278033 A092411 A272744

Adjacent sequences: A056494 A056495 A056496 * A056498 A056499 A056500

Keyword

nonn

Author

Marks R. Nester

Status

approved