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A059076 Number of pairs of orientable necklaces with n beads and two colors; i.e., turning the necklace over does not leave it unchanged. 12

%I

%S 0,0,0,0,0,0,1,2,6,14,30,62,128,252,495,968,1866,3600,6917,13286,

%T 25476,48916,93837,180314,346554,666996,1284570,2477342,4781502,

%U 9240012,17871708,34604066,67060746,130085052,252548760,490722344

%N Number of pairs of orientable necklaces with n beads and two colors; i.e., turning the necklace over does not leave it unchanged.

%C Number of chiral bracelets with n beads and two colors.

%H G. C. Greubel, <a href="/A059076/b059076.txt">Table of n, a(n) for n = 0..1000</a>

%H John P. McSorley and Alan H. Schoen, <a href="http://dx.doi.org/10.1016/j.disc.2012.08.021">Rhombic Tilings of (n,k)-Ovals, (n, k, lambda)-Cyclic Difference Sets, and Related Topics</a>, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012

%F a(n) = A000031(n) - A000029(n) = A000029(n) - A029744(n) = (A000031(n) - A029744(n))/2 = A008965(n) - A091696(n)

%F G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n - (1+x)^2/(1-2*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 + (1/2n)* Sum_{d|n} phi(d)*k^(n/d), where k=2 is the maximum number of colors. - _Robert A. Russell_, Sep 24 2018

%e For n=6, the only chiral pair is AABABB-AABBAB. For n=7, the two chiral pairs are AAABABB-AAABBAB and AABABBB-AABBBAB. - _Robert A. Russell_, Sep 24 2018

%t nn=35;Table[CoefficientList[Series[CycleIndex[CyclicGroup[n],s]-CycleIndex[DihedralGroup[n],s]/.Table[s[i]->2,{i,1,n}],{x,0,nn}],x],{n,1,nn}]//Flatten (* _Geoffrey Critzer_, Mar 26 2013 *)

%t mx=40; CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-2*x^n]/n, {n, mx}]-(1+x)^2/(1-2*x^2))/2, {x, 0, mx}], x] (* _Herbert Kociemba_, Nov 02 2016 *)

%t terms = 36; a29[0] = 1; a29[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#) & ]/(2*n); Array[a29, 36, 0] - LinearRecurrence[{0, 2}, {1, 2, 3}, 36] (* _Jean-Fran├žois Alcover_, Nov 05 2017 *)

%t k = 2; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n)(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}], 0] (* _Robert A. Russell_, Sep 24 2018 *)

%Y Column 2 of A293496.

%Y Cf. A059053.

%Y Column 2 of A305541.

%Y Equals (A000031 - A164090) / 2.

%Y a(n) = (A052823(n) - A027383(n-2)) / 2.

%K nonn

%O 0,8

%A _Henry Bottomley_, Dec 22 2000

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Last modified May 25 03:14 EDT 2019. Contains 323539 sequences. (Running on oeis4.)