%I #96 Feb 12 2021 01:43:06

%S 1,1,4,8,20,38,76,133,232,375,600,912,1368,1980,2829,3936,5412,7293,

%T 9724,12760,16588,21287,27092,34112,42640,52819,65008,79392,96405,

%U 116280,139536,166464,197676,233529,274740,321741,375364

%N Number of bracelets (turnover necklaces) of n beads of 2 colors, 7 of them black.

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

%C Also number of nonequivalent necklaces of 7 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=7 (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 Vincenzo Librandi, <a href="/A032280/b032280.txt">Table of n, a(n) for n = 7..1000</a>

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

%H S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, <a href="http://zfn.mpdl.mpg.de/data/Reihe_A/52/ZNA-1997-52a-0867.pdf">Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem</a>, Z. Naturforsch., 52a (1997), 867-873.

%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/0710.1370">A problem of enumeration of two-color bracelets with several variations</a>, arXiv:0710.1370 [math.CO], 2007-2011.

%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_16">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-8,6,6,-8,1,0,-1,8,-6,-6,8,0,-3,1).

%F S. J. Cyvin et al. (1997) give a g.f.

%F "DIK[ 7 ]" (necklace, indistinct, unlabeled, 7 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); 0, otherwise. Then

%F a(n) = (3/7)*s(n,0,7) + (48*C(n-1,6) + 7*(n-2)*(n-4)*(n-6))/672, if n is even;

%F a(n) = (3/7)*s(n,0,7) + (48*C(n-1,6) + 7*(n-1)*(n-3)*(n-5))/672, if n is odd. (End)

%F G.f.: -x^7*(4*x^6-2*x^5-2*x^4+4*x^3+x^2-2*x+1) / ((x-1)^7*(x+1)^3*(x^6+x^5+x^4+x^3+x^2+x+1)). - _Colin Barker_, Feb 06 2013

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

%F G.f.: (1/2)*x^7*((1+x)/(1-x^2)^4 + 1/7*(1/(1-x)^7 + 6/(1-x^7))).

%F G.f.: k=7, 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)

%t k = 7; 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 CoefficientList[Series[-(4 x^6 - 2 x^5 - 2 x^4 + 4 x^3 + x^2 - 2 x + 1)/((x - 1)^7 (x + 1)^3 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 19 2013 *)

%t k=7; 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=7 of A052307.

%K nonn,easy

%O 7,3

%A _Christian G. Bower_