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A032281 Number of bracelets (turnover necklaces) of n beads of 2 colors, 9 of them black.

%I A032281 #84 Apr 18 2025 05:49:59
%S A032281 1,1,5,12,35,79,185,375,750,1387,2494,4262,7105,11410,17930,27407,
%T A032281 41107,60335,87154,123695,173173,238957,325845,438945,585265,772252,
%U A032281 1009868,1308742,1682660,2146420,2718806,3419924,4274905,5310667,6560225,8059021,9849925
%N A032281 Number of bracelets (turnover necklaces) of n beads of 2 colors, 9 of them black.
%C A032281 From _Vladimir Shevelev_, Apr 23 2011: (Start)
%C A032281 Also number of non-equivalent necklaces of 9 beads each of them painted by one of n colors.
%C A032281 The sequence solves the so-called Reis problem about convex k-gons in the case k=9 (see our comment to A032279). (End)
%D A032281 N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.
%H A032281 Andrew Howroyd, <a href="/A032281/b032281.txt">Table of n, a(n) for n = 9..1000</a>
%H A032281 Christian G. Bower, <a href="/transforms2.html">Transforms (2)</a>.
%H A032281 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 A032281 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A032281 F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H A032281 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 A032281 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 A032281 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 A032281 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%H A032281 <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-6,-6,6,13,-2,-18,-1,11,3,0,0,-3,-11,1,18,2,-13,-6,6,6,-3,-2,1).
%F A032281 "DIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1...
%F A032281 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
%F A032281 From _Herbert Kociemba_, Nov 05 2016: (Start)
%F A032281 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))).
%F A032281 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)
%t A032281 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 *)
%t A032281 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 *)
%Y A032281 Column k=9 of A052307.
%K A032281 nonn,easy
%O A032281 9,3
%A A032281 _Christian G. Bower_