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A000046
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Number of primitive n-bead necklaces (turning over is allowed) where complements are equivalent.
(Formerly M0696 N0257)
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13
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1, 1, 1, 1, 2, 3, 5, 8, 14, 21, 39, 62, 112, 189, 352, 607, 1144, 2055, 3885, 7154, 13602, 25472, 48670, 92204, 176770, 337590, 649341, 1246840, 2404872, 4636389, 8964143, 17334800, 33587072, 65107998, 126387975, 245492232, 477349348
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OFFSET
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0,5
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COMMENTS
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Also, number of "twills" (Grünbaum and Shephard). - N. J. A. Sloane, Oct 21 2015
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REFERENCES
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B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-98 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = Sum_{ d divides n } mu(d)*A000011(n/d).
a(n) = ((1/(2n))Sum_{odd d|n} mu(d)*2^(n/d) + Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2.
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EXAMPLE
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For a(7)=8, there are seven achiral set partitions (0000001, 0000011, 0000101, 0000111, 0001001, 0010011, 0010101) and one chiral pair (0001011-0001101). - Robert A. Russell, Jun 19 2019
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MAPLE
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with(numtheory); A000046 := proc(n) local s, d; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*A000011(n/d); od; RETURN(s); fi; end;
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MATHEMATICA
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a11[0] = 1; a11[n_] := 2^Floor[n/2]/2 + Sum[EulerPhi[2*d]*2^(n/d), {d, Divisors[n]}]/n/4; a[0] = 1; a[n_] := Sum[MoebiusMu[d]*a11[n/d], {d, Divisors[n]}]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jul 10 2012, from formula *)
Join[{1}, Table[(DivisorSum[NestWhile[#/2 &, n, EvenQ], MoebiusMu[#] 2^(n/#) &]/(2 n) + DivisorSum[n, MoebiusMu[n/#] 2^Floor[#/2] &])/2, {n, 1, 40}]] (* Robert A. Russell, Jun 19 2019 *)
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CROSSREFS
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Similar to A000011, but counts primitive necklaces.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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