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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
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
KEYWORD
nonn
AUTHOR
Henry Bottomley, Dec 22 2000
STATUS
approved

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Last modified June 23 06:39 EDT 2024. Contains 373629 sequences. (Running on oeis4.)