A032282 Number of bracelets (turnover necklaces) of n beads of 2 colors, 11 of them black.

1, 1, 6, 16, 56, 147, 392, 912, 2052, 4262, 8524, 16159, 29624, 52234, 89544, 148976, 242086, 384111, 597506, 911456, 1367184, 2017509, 2934559, 4209504, 5963464, 8347612, 11558232, 15837472, 21493712, 28903332

Offset

11,3

Comments

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of non-equivalent necklaces of 11 beads each of them painted by one of n colors.

The sequence solves the so-called Reis problem about convex k-gons in case k=11 (see our comment to A032279).

(End)

References

N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

Links

Andrew Howroyd, Table of n, a(n) for n = 11..1000

Christian G. Bower, Transforms (2).

Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.

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]

Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.

Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.

Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).

Index entries for sequences related to bracelets

Index entries for linear recurrences with constant coefficients, signature (5,-5,-15,35,1,-65,45,45,-65,1,36,-20,0,20,-36,-1,65,-45,-45,65,-1,-35,15,5,-5,1).

Formula

"DIK[ 11 ]" (necklace, indistinct, unlabeled, 11 parts) transform of 1, 1, 1, 1...

From Vladimir Shevelev, Apr 23 2011: (Start)

Put s(n,k,d)=1, if n==k(mod d), and s(n,k,d)=0, otherwise. Then

a(n) = 5*s(n,0,11)/11+(3840*C(n-1,10)+11*(n-2)*(n-4)*(n-6)(n-8)*(n-10))/84480, if n is even;

a(n) = 5*s(n,0,11)/11+(3840*C(n-1,10)+11*(n-1)*(n-3)*(n-5)*(n-7)*(n-9))/84480, if n is odd.

(End)

From Herbert Kociemba, Nov 05 2016: (Start)

G.f.: 1/22*x^11*(1/(1-x)^11 + 11/((-1+x)^6*(1+x)^5) - 10/(-1+x^11)).

G.f.: k=11, x^k*((1/k)*Sum_{d|k} phi(d)*(1-x^d)^(-k/d) + (1+x)/(1-x^2)^floor[(k+2)/2])/2. [edited by Petros Hadjicostas, Jul 18 2018]

(End)

Mathematica

k = 11; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
k=11; CoefficientList[Series[x^k*(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2, {x, 0, 50}], x] (* Herbert Kociemba, Nov 04 2016 *)

Crossrefs

Column k=11 of A052307.

Sequence in context: A221270 A316984 A192000 * A084057 A163302 A223028

Adjacent sequences: A032279 A032280 A032281 * A032283 A032284 A032285

Keyword

nonn

Author

Christian G. Bower

Status

approved