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 A032280 Number of bracelets (turnover necklaces) of n beads of 2 colors, 7 of them black. 6
 1, 1, 4, 8, 20, 38, 76, 133, 232, 375, 600, 912, 1368, 1980, 2829, 3936, 5412, 7293, 9724, 12760, 16588, 21287, 27092, 34112, 42640, 52819, 65008, 79392, 96405, 116280, 139536, 166464, 197676, 233529, 274740, 321741, 375364 (list; graph; refs; listen; history; text; internal format)
 OFFSET 7,3 COMMENTS From Vladimir Shevelev, Apr 23 2011: (Start) Also number of nonequivalent necklaces of 7 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=7 (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 Vincenzo Librandi, Table of n, a(n) for n = 7..1000 C. G. Bower, Transforms (2) S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal systems including the corannulene and coronene homologs: novel applications of PĆ³lya's theorem, Z. Naturforsch., 52a (1997), 867-873. 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. [Cached copy, with permission, pdf format only] V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638. V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011. 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 linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,1,0,-1,8,-6,-6,8,0,-3,1). FORMULA S. J. Cyvin et al. (1997) give a g.f. "DIK[ 7 ]" (necklace, indistinct, unlabeled, 7 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); 0, otherwise. Then 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; 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) 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 From Herbert Kociemba, Nov 05 2016: (Start) G.f.: 1/2*x^7*((1+x)/(1-x^2)^4+1/7*(1/(1-x)^7+6/(1-x^7))). 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) MATHEMATICA 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 *) 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 *) 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 *) CROSSREFS Sequence in context: A133628 A280486 A097940 * A300158 A156303 A301138 Adjacent sequences:  A032277 A032278 A032279 * A032281 A032282 A032283 KEYWORD nonn,easy,changed AUTHOR STATUS approved

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Last modified January 22 20:23 EST 2019. Contains 319365 sequences. (Running on oeis4.)