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A056289
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Number of primitive (period n) n-bead necklaces with exactly four different colored beads.
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5
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0, 0, 0, 6, 48, 260, 1200, 5100, 20720, 81828, 318000, 1222870, 4675440, 17813820, 67769504, 257695800, 980240880, 3731732200, 14222737200, 54278498154, 207438936800, 793940157900, 3043140078000, 11681056021300, 44900438149248, 172824327151140, 666070256468960
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OFFSET
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1,4
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COMMENTS
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Turning over the necklace is not allowed.
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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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Alois P. Heinz, Table of n, a(n) for n = 1..1000
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FORMULA
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a(n) = Sum_{d|n} mu(d)*A056284(n/d).
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MAPLE
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with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 4-j)*binomial(4, j)*(-1)^j, j=0..4):
seq(a(n), n=1..30); # Alois P. Heinz, Jan 25 2015
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MATHEMATICA
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b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^#&]/n]; a[n_] := Sum[b[n, 4-j]*Binomial[4, j]*(-1)^j, {j, 0, 4}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 20 2017, after Alois P. Heinz *)
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CROSSREFS
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Cf. A027377.
Column k=4 of A254040.
Sequence in context: A353247 A262354 A052771 * A056284 A293967 A246587
Adjacent sequences: A056286 A056287 A056288 * A056290 A056291 A056292
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KEYWORD
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nonn
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AUTHOR
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Marks R. Nester
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STATUS
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approved
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