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
KEYWORD
nonn
AUTHOR
Henry Bottomley, Dec 22 2000
STATUS
approved