A056496 Number of primitive (period n) periodic palindromes using a maximum of five different symbols.

5, 10, 20, 60, 120, 340, 620, 1800, 3100, 9240, 15620, 46440, 78120, 233740, 390480, 1170000, 1953120, 5855900, 9765620, 29287440, 48827480, 146468740, 244140620, 732373200, 1220703000, 3662031240

Offset

1,1

Comments

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

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..26.

Formula

a(n) = Sum_{d|n} mu(d)*A056487(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, 5], {x, 0, mx}], x] (* Herbert Kociemba, Nov 29 2016 *)

Crossrefs

Column 5 of A284856.

Cf. A056461.

Sequence in context: A023383 A229171 A092407 * A106367 A244026 A244022

Adjacent sequences: A056493 A056494 A056495 * A056497 A056498 A056499

Keyword

nonn

Author

Marks R. Nester

Status

approved