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A056493 Number of primitive (period n) periodic palindromes using a maximum of two different symbols. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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..42.

Index entries for sequences related to Lyndon words

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.)

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)

G.f.: Sum_{n>=1} mu(n)*x^n*(2+3*x^n)/(1-2*x^(2*n)). The g.f. by H. 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

EXAMPLE

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

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 *)

CROSSREFS

Column 2 of A284856.

Cf. A056458.

Sequence in context: A108618 A097719 A249050 * A289352 A277619 A001371

Adjacent sequences:  A056490 A056491 A056492 * A056494 A056495 A056496

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

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

STATUS

approved

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Last modified November 21 15:35 EST 2017. Contains 295002 sequences.