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A032275 Number of bracelets (turn over necklaces) of n beads of 4 colors. 9
4, 10, 20, 55, 136, 430, 1300, 4435, 15084, 53764, 192700, 704370, 2589304, 9608050, 35824240, 134301715, 505421344, 1909209550, 7234153420, 27489127708, 104717491064, 399827748310, 1529763696820 (list; graph; refs; listen; history; text; internal format)
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.

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 + (1/2n)* Sum_{d|n} phi(d)*k^(n/d), where k=4 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

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,changed

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified October 19 02:23 EDT 2018. Contains 316327 sequences. (Running on oeis4.)