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 A000046 Number of primitive n-bead necklaces (turning over is allowed) where complements are equivalent. (Formerly M0696 N0257) 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also, number of "twills" (Grünbaum and Shephard). - N. J. A. Sloane, Oct 21 2015 REFERENCES 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). LINKS Christian G. Bower, Table of n, a(n) for n = 0..1000 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32. FORMULA a(n) = Sum_{ d divides n } mu(d)*A000011(n/d). From Robert A. Russell, Jun 19 2019: (Start) a(n) = ((1/(2n))Sum_{odd d|n} mu(d)*2^(n/d) + Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2. a(n) = A000048(n) - A308706(n) = (A000048(n) + A179781(n))/2 = A308706(n) + A179781(n). A000011(n) =  Sum_{d|n} a(d). (End) EXAMPLE 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 MAPLE 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; MATHEMATICA 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 *) CROSSREFS Similar to A000011, but counts primitive necklaces. A000048 (oriented), A308706 (chiral), A179781 (achiral). Cf. A054199. Sequence in context: A306912 A212607 A056366 * A293641 A293553 A131132 Adjacent sequences:  A000043 A000044 A000045 * A000047 A000048 A000049 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)