A056495 Number of primitive (period n) periodic palindromes using a maximum of four different symbols.

4, 6, 12, 30, 60, 138, 252, 600, 1008, 2490, 4092, 10050, 16380, 40698, 65460, 163200, 262140, 654192, 1048572, 2618850, 4194036, 10481658, 16777212, 41932200, 67108800, 167755770, 268434432, 671047650

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]

Links

Table of n, a(n) for n=1..28.

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

Column 4 of A284856.

Cf. A056460.

Sequence in context: A126259 A068570 A092320 * A351523 A263656 A178674

Adjacent sequences: A056492 A056493 A056494 * A056496 A056497 A056498

Keyword

nonn

Author

Marks R. Nester

Status

approved