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A056496
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Number of primitive (period n) periodic palindromes using a maximum of five different symbols.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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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)*A056487(n/d).
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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EXAMPLE
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For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
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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, 5], {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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