A056493 Number of primitive (period n) periodic palindromes using a maximum of two different symbols.

2, 1, 2, 3, 6, 7, 14, 18, 28, 39, 62, 81, 126, 175, 246, 360, 510, 728, 1022, 1485, 2030, 3007, 4094, 6030, 8184, 12159, 16352, 24381, 32766, 48849, 65534, 97920, 131006, 196095, 262122, 392364, 524286, 785407, 1048446, 1571310, 2097150, 3143497

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

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

Also number of aperiodic necklaces (Lyndon words) with two colors that are the same when turned over.

Examples

			a(1) = 2 with aaa... and bbb..., a(2) = 1 with ababab..., a(3) = 2 with aabaab... and abbabb..., a(4) = 3 with aaabaaab... and aabbaabb... and abbbabbb.... - _Michael Somos_, 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 a pdf file of Chap. 2]

Links

• Index entries for sequences related to Lyndon words

Crossrefs

Column 2 of A284856.

Cf. A029744, A056391, A056458.

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

Formula

Sum_{d|n} mu(d)*b(n/d), where b(n) = A029744(n+1). [Corrected by Petros Hadjicostas, Oct 15 2017. The original formula referred to a previous version of sequence A029744 that had a different offset.]

More generally, let gf(k) be the g.f. for the number of necklaces with reflectional symmetry but no rotational symmetry and beads of k colors. Then gf(k): Sum_{n >= 1} mu(n)*Sum_{i=0..2} binomial(k,i)*x^(n*i)/(1 - k*x^(2*n)). - Herbert Kociemba, Nov 29 2016

G.f.: Sum_{n >= 1} mu(n)*x^n*(2 + 3*x^n)/(1 - 2*x^(2*n)). The g.f. by Herbet Kociemba above, with k = 2, becomes Sum_{n>=1} mu(n)*(x^n + 1)^2/(1 - 2*x^(2*n)). The two formulae differ by the "undetermined" constant Sum_{n >= 1} mu(n). - Petros Hadjicostas, Oct 15 2017

Extensions

More terms and additional comments from Christian G. Bower, Jun 22 2000