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 A001371 Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is allowed. (Formerly M0115 N0045 N0285) 9
 1, 2, 1, 2, 3, 6, 8, 16, 24, 42, 69, 124, 208, 378, 668, 1214, 2220, 4110, 7630, 14308, 26931, 50944, 96782, 184408, 352450, 675180, 1296477, 2493680, 4805388, 9272778, 17919558, 34669600, 67156800, 130215996, 252741255, 490984464, 954629662, 1857545298 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence, in two entries, N0045 and N0285). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..400 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only] FORMULA a(n) = Sum_{ d divides n } mu(d)*A000029(n/d). From Herbert Kociemba, Nov 28 2016: (Start) More generally, for n>0, gf(k) is the g.f. for the number of bracelets with primitive period n and beads of k colors. gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n + Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End) MAPLE with(numtheory); A001371 := proc(n) local s, d; if n = 0 then RETURN(1) else s := 0; for d in divisors(n) do s := s+mobius(d)*A000029(n/d); od; RETURN(s); fi; end; MATHEMATICA a29[n_] := a29[n] = (s = If[OddQ[n], 2^((n-1)/2) , 2^(n/2 - 2) + 2^(n/2 - 1)]; a29[0] = 1; Do[s = s + EulerPhi[d]*2^(n/d)/(2*n), {d, Divisors[n]}]; s); a[n_] := Sum[ MoebiusMu[d]*a29[n/d], {d, Divisors[n]}]; a[0] = 1; Table[ a[n], {n, 0, 34}] (* Jean-François Alcover, Oct 04 2011 *) mx=40; gf[x_, k_]:=Sum[ MoebiusMu[n]*(-Log[1-k*x^n]/n+Sum[Binomial[k, i]x^(n i), {i, 0, 2}]/( 1-k x^(2n)))/2, {n, mx}]; ReplacePart[CoefficientList[Series[gf[x, 2], {x, 0, mx}], x], 1->1] (* Herbert Kociemba, Nov 28 2016 *) (Python) from sympy import divisors, totient, mobius def a000029(n): return 1 if n<1 else (n%2 + 3)/4*2**int(n/2) + sum([totient(n/d)*2**d for d in divisors(n)])/(2*n) def a(n): return 1 if n<1 else sum([mobius(d)*a000029(n/d) for d in divisors(n)]) print [a(n) for n in xrange(51)] # Indranil Ghosh, Apr 23 2017 CROSSREFS Column 2 of A276550. Sequence in context: A056493 A289352 A277619 * A277629 A277631 A277633 Adjacent sequences:  A001368 A001369 A001370 * A001372 A001373 A001374 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Christian G. Bower Entry revised by N. J. A. Sloane, Jun 10 2012 STATUS approved

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Last modified January 17 18:48 EST 2019. Contains 319251 sequences. (Running on oeis4.)