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%I M4105 #72 Feb 11 2021 09:22:22
%S 1,1,6,14,47,111,280,600,1282,2494,4752,8524,14938,25102,41272,65772,
%T 102817,156871,235378,346346,502303,716859,1010256,1404624,1931540,
%U 2625658,3534776,4711448,6226148,8156396,10603704
%N Number of n-bead bracelets (turnover necklaces) of two colors with 10 red beads and n-10 black beads.
%C From _Vladimir Shevelev_, Apr 23 2011: (Start)
%C Also number of non-equivalent (turnover) necklaces of 10 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=10 (see our comment to A032279). (End)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%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="/A005515/b005515.txt">Table of n, a(n) for n = 10..1000</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 W. D. Hoskins and Anne Penfold Street, <a href="https://dx.doi.org/10.1017/S1446788700017547">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. Ser. A, 33 (1982), no. 1, 1-15.
%H W. D. Hoskins and A. P. Street, <a href="/A005513/a005513_1.pdf">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
%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. P. Street, <a href="/A005513/a005513.pdf">Letter to N. J. A. Sloane, N.D.</a>
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%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 a(n) = n*s(n,0,5)/25 + ((384*C(n-1,9) + (n+1)*(n-2)*(n-4)*(n-6)*(n-8))/7680, if n is even; a(n) = (n-5)*s(n,0,5)/25 + ((384*C(n-1,9) + (n-1)*(n-3)*(n-5)*(n-7)*(n-9))/7680, if n is odd. (End)
%F From _Herbert Kociemba_, Nov 04 2016: (Start)
%F G.f.: (1/20)*x^10*(1/(-1+x)^10 + 10/((-1+x)^6*(1+x)^5) + 1/(1-x^2)^5 + 4/(-1+x^5)^2 - 4/(-1+x^10)).
%F G.f.: k=10, 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_, Jan 10 2019] (End)
%t k = 10; 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=10;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=10 of A052307.
%K nonn
%O 10,3
%A _N. J. A. Sloane_
%E Sequence extended and description corrected by _Christian G. Bower_
%E Name edited by _Petros Hadjicostas_, Jan 10 2019
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