A278642 Number of pairs of orientable necklaces with n beads and up to 6 colors; i.e., turning the necklace over does not leave it unchanged. The turned-over necklace is not included in the count.

0, 0, 0, 20, 105, 672, 3535, 19350, 102795, 556010, 3010098, 16467450, 90619690, 502194420, 2798240265, 15671993560, 88156797855, 497837886000, 2821092554035, 16035752398770, 91403856697944, 522308167195260, 2991401733402075, 17168047238861070, 98716274117752900, 568605754068247644, 3280417827002225910, 18953525314104758810

Offset

0,4

Comments

Number of chiral bracelets of n beads using up to six different colors.

Links

Table of n, a(n) for n=0..27.

Formula

Equals (A054625(n) - A056488(n)) / 2 = A054625(n) - A056341(n) = A056341(n) - A056488(n), for n >= 1.

G.f.: k = 6, (1 - Sum_{n >= 1} phi(n)*log(1 - k*x^n)/n - Sum_{i = 0..2} Binomial[k, i]*x^i / ( 1 - k*x^2) )/2.

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 = 6 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

Mathematica

mx = 40; f[x_, k_] := (1 - Sum[EulerPhi[n] * Log[1 - k * x^n]/n, {n, mx}] - Sum[Binomial[k, i] * x^i, {i, 0, 2}]/(1 - k * x^2))/2; CoefficientList[Series[f[x, 6], {x, 0, mx}], x]
k = 6; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) - (k^Floor[(n + 1)/2] + k^Ceiling[(n + 1)/2])/4, {n, 30}], 0] (* Robert A. Russell, Sep 24 2018 *)

Crossrefs

Column 6 of A293496.

Cf. A059076 (2 colors), A278639 (3 colors), A278640 (4 colors), A278641 (5 colors).

Sequence in context: A187756 A248087 A209547 * A135174 A173963 A202957

Adjacent sequences: A278639 A278640 A278641 * A278643 A278644 A278645

Keyword

nonn

Author

Herbert Kociemba, Nov 24 2016

Status

approved