%I #71 Feb 11 2021 09:02:09

%S 1,1,6,16,56,147,392,912,2052,4262,8524,16159,29624,52234,89544,

%T 148976,242086,384111,597506,911456,1367184,2017509,2934559,4209504,

%U 5963464,8347612,11558232,15837472,21493712,28903332

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

%C From _Vladimir Shevelev_, Apr 23 2011: (Start)

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

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

%C (End)

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

%H Andrew Howroyd, <a href="/A032282/b032282.txt">Table of n, a(n) for n = 11..1000</a>

%H Christian G. Bower, <a href="/transforms2.html">Transforms (2)</a>.

%H Hansraj Gupta, <a href="https://web.archive.org/web/20200806162943/https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a66_964.pdf">Enumeration of incongruent cyclic k-gons</a>, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.

%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]

%H Vladimir Shevelev, <a href="https://web.archive.org/web/20200722171019/http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/2000c4e8_629.pdf">Necklaces and convex k-gons</a>, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.

%H Vladimir Shevelev, <a href="https://www.math.bgu.ac.il/~shevelev/Shevelev_Neclaces.pdf">Necklaces and convex k-gons</a>, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1104.4051">Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma)</a>, arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).

%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>

%H <a href="/index/Rec#order_26">Index entries for linear recurrences with constant coefficients</a>, 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).

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

%F From _Vladimir Shevelev_, Apr 23 2011: (Start)

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

%F 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;

%F 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.

%F (End)

%F From _Herbert Kociemba_, Nov 05 2016: (Start)

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

%F 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]

%F (End)

%t 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 *)

%t 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 *)

%Y Column k=11 of A052307.

%K nonn

%O 11,3

%A _Christian G. Bower_