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%I #87 May 05 2021 00:18:30
%S 1,3,6,10,21,39,92,198,498,1219,3210,8418,22913,62415,173088,481598,
%T 1351983,3808083,10781954,30615354,87230157,249144711,713387076,
%U 2046856566,5884491500,16946569371,48883660146,141217160458,408519019449,1183289542815
%N Number of necklaces with n beads of 3 colors, allowing turning over.
%C Number of bracelets of n beads using up to three different colors. - _Robert A. Russell_, Sep 24 2018
%D J. L. Fisher, Application-Oriented Algebra (1977), ISBN 0-7002-2504-8, circa p. 215.
%D M. Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pp. 245-246.
%H T. D. Noe, <a href="/A027671/b027671.txt">Table of n, a(n) for n = 0..200</a>
%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H M. Taniguchi, H. Du, and J. S. Lindsey, <a href="https://doi.org/10.1021/ci400175f">Enumeration of virtual libraries of combinatorial modular macrocyclic (bracelet, necklace) architectures and their linear counterparts</a>, Journal of Chemical Information and Modeling, 53 (2013), 2203-2216.
%H R. M. Thompson and R. T. Downs, <a href="http://www.geo.arizona.edu/xtal//group/pdf/acB57766.pdf">Systematic generation of all nonequivalent closest-packed stacking sequences of length N using group theory</a>, Acta Cryst. B57 (2001), 766-771; B58 (2002), 153.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Necklace.html">Necklace.</a>
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 3*x^n)/n + (1+3*x+3*x^2)/(1-3*x^2))/2. - _Herbert Kociemba_, Nov 02 2016
%F For n > 0, a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))* Sum_{d|n} phi(d)*k^(n/d), where k=3 is the maximum number of colors. - _Robert A. Russell_, Sep 24 2018
%F a(0) = 1; a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i), where k=3 is the maximum number of colors.
%F (See A075195 formulas.) - _Richard L. Ollerton_, May 04 2021
%e For n=2, the six bracelets are AA, AB, AC, BB, BC, and CC. - _Robert A. Russell_, Sep 24 2018
%t Needs["Combinatorica`"]; Join[{1}, Table[CycleIndex[DihedralGroup[n], s]/.Table[s[i]->3, {i,1,n}], {n,1,30}]] (* _Geoffrey Critzer_, Sep 29 2012 *)
%t Needs["Combinatorica`"]; Join[{1}, Table[NumberOfNecklaces[n, 3, Dihedral], {n, 30}]] (* _T. D. Noe_, Oct 02 2012 *)
%t mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-3*x^n]/n,{n,mx}]+(1+3 x+3 x^2)/(1-3 x^2))/2,{x,0,mx}],x] (* _Herbert Kociemba_, Nov 02 2016 *)
%t t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1+k)*k^(n/2))/(2*n), (t1 + n*k^((n+1)/2))/(2*n)]); a[0] = 1; a[n_] := t[n, 3]; Array[a, 30, 0] (* _Jean-François Alcover_, Nov 02 2017, after Maple code for A081720 *)
%t k=3; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}], 1] (* _Robert A. Russell_, Sep 24 2018 *)
%o (PARI) a(n,k=3) = if(n==0,1,(k^floor((n+1)/2) + k^ceil((n+1)/2))/4 + (1/(2*n))* sumdiv(n, d, eulerphi(d)*k^(n/d) ) );
%o vector(55,n,a(n-1)) \\ _Joerg Arndt_, Oct 20 2019
%Y Cf. A056353, A114438.
%Y a(n) = A081720(n,3), n >= 3. - _Wolfdieter Lang_, Jun 03 2012
%Y Column 3 of A051137.
%Y a(n) = (A001867(n) + A182751(n+1)) / 2 = A278639(n) + A182751(n+1).
%Y Equals A001867 - A278639.
%K nonn,easy,nice
%O 0,2
%A _Alford Arnold_
%E More terms from _Christian G. Bower_
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