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A056494
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Number of primitive (period n) periodic palindromes using a maximum of three different symbols.
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1
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3, 3, 6, 12, 24, 42, 78, 144, 234, 456, 726, 1392, 2184, 4290, 6528, 12960, 19680, 39078, 59046, 117600, 177060, 353562, 531438, 1061280, 1594296, 3186456, 4782726, 9561552, 14348904, 28690752, 43046718
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OFFSET
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1,1
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COMMENTS
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For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
Number of aperiodic necklaces with three colors that are the same when turned over and hence have reflectional symmetry but no rotational symmetry. - Herbert Kociemba, Nov 29 2016
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REFERENCES
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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]
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LINKS
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FORMULA
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a(n) = Sum_{d|n} mu(d)*A038754(n/d+1).
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)
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MATHEMATICA
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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, 3], {x, 0, mx}], x] (* Herbert Kociemba, Nov 29 2016 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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