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 A032281 Number of bracelets (turnover necklaces) of n beads of 2 colors, 9 of them black. 5
 1, 1, 5, 12, 35, 79, 185, 375, 750, 1387, 2494, 4262, 7105, 11410, 17930, 27407, 41107, 60335, 87154, 123695, 173173, 238957, 325845, 438945, 585265, 772252, 1009868, 1308742, 1682660, 2146420, 2718806, 3419924, 4274905 (list; graph; refs; listen; history; text; internal format)
 OFFSET 9,3 COMMENTS From Vladimir Shevelev, Apr 23 2011: (Start) Also number of non-equivalent necklaces of 9 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=9 (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 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. [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, 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). FORMULA "DIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1... Put s(n,k,d)=1, if n==k (mod d), and s(n,k,d)=0, otherwise. Then a(n)=(1/3)*s(n,0,9)+(n-3)*(n-6)*s(n,0,3)/162+(n-2)(n-4)*(n-6)*(n-8)*(945+(n-1)*(n-3)*(n-5)*(n-7))/725760, if n is even; a(n)=(1/3)*s(n,0,9)+(n-3)*(n-6)*s(n,0,3)/162+(n-1)*(n-3)*(n-5)*(n-7)*(945+(n-2)*(n-4)*(n-6)*(n-8))/725760, if n is odd. - Vladimir Shevelev, Apr 23 2011 From Herbert Kociemba, Nov 05 2016: (Start) G.f.: 1/2*x^9*((1+x)/(1-x^2)^5 + 1/9*(1/(1-x)^9 - 2/(-1+x^3)^3 - 6/(-1+x^9))). G.f.: k=9, 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 = 9; 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=9; 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: A298992 A050189 A116995 * A294654 A229043 A185699 Adjacent sequences:  A032278 A032279 A032280 * A032282 A032283 A032284 KEYWORD nonn,changed AUTHOR STATUS approved

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)