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

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

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

Daniel Gabric and Joe Sawada, Efficient Construction of Long Orientable Sequences, arXiv:2401.14341 [cs.DS], 2024.

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-François 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 A343889

Adjacent sequences: A059073 A059074 A059075 * A059077 A059078 A059079

Keyword

nonn

Author

Henry Bottomley, Dec 22 2000

Status

approved