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A032282 Number of bracelets (turn over necklaces) of n beads of 2 colors, 11 of them black. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

LINKS

Table of n, a(n) for n=11..40.

C. G. Bower, Transforms (2)

H. 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.

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

V. 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

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),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*Plus@@(EulerPhi[#]*(1-x^#)^(-(k/#))&/@Divisors[k]) + (1+x)/(1-x^2)^Floor[(k+2)/2])/2. (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

Sequence in context: A175659 A221270 A192000 * A084057 A163302 A223028

Adjacent sequences:  A032279 A032280 A032281 * A032283 A032284 A032285

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified February 22 09:36 EST 2018. Contains 299448 sequences. (Running on oeis4.)