A032275 Number of bracelets (turnover necklaces) of n beads of 4 colors.

4, 10, 20, 55, 136, 430, 1300, 4435, 15084, 53764, 192700, 704370, 2589304, 9608050, 35824240, 134301715, 505421344, 1909209550, 7234153420, 27489127708, 104717491064, 399827748310, 1529763696820

Offset

1,1

Links

Table of n, a(n) for n=1..23.

C. G. Bower, Transforms (2)

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

M. Taniguchi, H. Du, and J. S. Lindsey, Enumeration of virtual libraries of combinatorial modular macrocyclic (bracelet, necklace) architectures and their linear counterparts, Journal of Chemical Information and Modeling, 53 (2013), 2203-2216.

Index entries for sequences related to bracelets

Formula

"DIK" (bracelet, indistinct, unlabeled) transform of 4, 0, 0, 0, ...

Equals (A001868(n) + A056486(n)) / 2 = A001868(n) - A278640(n) = A278640(n) + A056486(n), for n>=1.

a(n) = A081720(n,4), n >= 4. - Wolfdieter Lang, Jun 03 2012

G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 4*x^n)/n + (1+4*x+6*x^2)/(1-4*x^2))/2. - Herbert Kociemba, Nov 02 2016

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=4 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i), where k=4 is the maximum number of colors. (See A075195 formulas.) - Richard L. Ollerton, May 04 2021

Example

For n=2, the ten bracelets are AA, AB, AC, AD, BB, BC, BD, CC, CD, and DD. - Robert A. Russell, Sep 24 2018

Mathematica

mx=40; CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-4*x^n]/n, {n, mx}]+(1+4 x+6 x^2)/(1-4 x^2))/2, {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
k=4; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* Robert A. Russell, Sep 24 2018 *)

Crossrefs

Column 4 of A051137.

Sequence in context: A237626 A020149 A056412 * A220828 A015220 A047199

Adjacent sequences: A032272 A032273 A032274 * A032276 A032277 A032278

Keyword

nonn

Author

Christian G. Bower

Status

approved