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A056391
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Number of step shifted (decimated) sequence structures using a maximum of two different symbols.
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293
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1, 2, 3, 6, 6, 20, 14, 48, 52, 140, 108, 624, 352, 1400, 2172, 4464, 4116, 22112, 14602, 68016, 88376, 209936, 190746, 1075200, 839128, 2797000, 3730584, 11276704, 9587580, 67195520, 35792568
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OFFSET
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1,2
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COMMENTS
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See A056371 for an explanation of step shifts. Permuting the symbols will not change the structure.
Also, number of circulant digraphs on n vertices up to Cayley isomorphism. Two circulant graphs are Cayley isomorphic if there is a d, which is necessarily prime to n, that transforms through multiplication modulo n the step values of one graph into those of the other. For squarefree n this is the only way that two circulant graphs can be isomorphic (see A049297). - Andrew Howroyd, Apr 20 2017
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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.
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LINKS
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FORMULA
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Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
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MATHEMATICA
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a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; a[n_] := a[2, n]/2; Array[a, 40] (* Jean-François Alcover, Jun 12 2017 *)
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PROG
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(PARI) a(n)=sum(k=1, n, if(gcd(k, n)==1, 2^(sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d)))-1), 0))/eulerphi(n); \\ Andrew Howroyd, Apr 20 2017
(PARI) \\ alternative using Polya enumeration functions (see attachment)
a(n) = NonequivalentStructs(StepShiftPerms(n), 2); \\ Andrew Howroyd, Oct 01 2017
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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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