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A032276 Number of bracelets (turnover necklaces) with n beads of 5 colors. 6
5, 15, 35, 120, 377, 1505, 5895, 25395, 110085, 493131, 2227275, 10196680, 46989185, 218102685, 1017448143, 4768969770, 22440372245, 105966797755, 501938733555, 2384200683816, 11353290089305 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Petros Hadjicostas, Sep 01 2018: (Start)

The DIK transform of the sequence (c(n): n >= 1), with g.f. C(x) = Sum_{n >= 1} c(n)*x^n, has g.f. -(1/2)*Sum_{m >= 1} (phi(m)/m))*log(1-C(x^m)) + (1 + C(x))^2/(4*(1-C(x^2))) - 1/4.

Here, c(1) = 5 and c(n) = 0 for n >= 2, and thus, C(x) = 5*x. Substituting this to the above g.f., we get that the g.f. of the current sequence is A(x) = Sum_{n >= 1} a(n)*x^n = -(1/2)*Sum_{m >= 1} (phi(m)/m))*log(1-5*x^m) + (1 + 5*x)^2/(4*(1-5*x^2)) - 1/4. This agrees with Herbert Kociemba's g.f. below except for an extra 1 because (1 + (1+5*x+10*x^2)/(1-5*x^2))/2 = 1 + (1 + 5*x)^2/(4*(1-5*x^2)) - 1/4.

(End)

LINKS

Table of n, a(n) for n=1..21.

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 5, 0, 0, 0, ...

a(n) = A081720(n,5), n >= 1. - Wolfdieter Lang, Jun 03 2012

G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 5*x^n)/n + (1+5*x+10*x^2)/(1-5*x^2))/2. - Herbert Kociemba, Nov 02 2016

a(n) = (3/2)*5^(n/2) + (1/(2*n))*Sum_{d|n} phi(n/d)*5^d, if n is even, and = (1/2)*5^((n+1)/2) + (1/(2*n))*Sum_{d|n} phi(n/d)*5^d, if n is odd. - Petros Hadjicostas, Sep 01 2018

a(n) = (A001869(n) + A056487(n+1)) / 2 = A278641(n) + A056487(n+1) = A001869(n) - A278641(n). - Robert A. Russell, Oct 13 2018

EXAMPLE

For n=2, the 15 bracelets are AA, AB, AC, AD, AE, BB, BC, BD, BE, CC, CD, CE, DD, DE, and EE. - Robert A. Russell, Sep 24 2018

MATHEMATICA

mx=40; CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-5*x^n]/n, {n, mx}]+(1+5 x+10 x^2)/(1-5 x^2))/2, {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)

k=5; 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

Cf. A081720.

Column 5 of A051137.

Cf. A001869 (oriented), A056487 (achiral), A278641 (chiral).

Sequence in context: A292912 A091875 A056413 * A065780 A220480 A105720

Adjacent sequences:  A032273 A032274 A032275 * A032277 A032278 A032279

KEYWORD

nonn,changed

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified October 17 22:48 EDT 2018. Contains 316297 sequences. (Running on oeis4.)