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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
0, 0, 0, 0, 0, 0, 1, 2, 6, 14, 30, 62, 128, 252, 495, 968, 1866, 3600, 6917, 13286, 25476, 48916, 93837, 180314, 346554, 666996, 1284570, 2477342, 4781502, 9240012, 17871708, 34604066, 67060746, 130085052, 252548760, 490722344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Number of chiral bracelets with n beads and two colors.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Petros Hadjicostas, Formulas for chiral bracelets, 2019; see Section 5.

John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From N. J. A. Sloane, Nov 26 2012

FORMULA

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

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

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 = 2 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

EXAMPLE

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

MATHEMATICA

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 *)

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 *)

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 *)

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 *)

CROSSREFS

Column 2 of A293496.

Cf. A059053.

Column 2 of A305541.

Equals (A000031 - A164090) / 2.

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

Sequence in context: A095121 A296965 A000918 * A237623 A232059 A263711

Adjacent sequences:  A059073 A059074 A059075 * A059077 A059078 A059079

KEYWORD

nonn

AUTHOR

Henry Bottomley, Dec 22 2000

STATUS

approved

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Last modified October 18 10:05 EDT 2019. Contains 328146 sequences. (Running on oeis4.)