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A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads.

%I A005516 M4368 #73 Feb 26 2025 11:22:14
%S A005516 1,1,7,19,72,196,561,1368,3260,7105,14938,29624,56822,104468,186616,
%T A005516 322786,544802,896259,1444147,2278640,3532144,5380034,8070400,
%U A005516 11926928,17393969,25042836,35638596,50152013,69855536
%N A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads.
%C A005516 From _Vladimir Shevelev_, Apr 23 2011: (Start)
%C A005516 Also number of non-equivalent (turnover) necklaces of 12 beads each of them painted by one of n colors.
%C A005516 The sequence solves the so-called Reis problem about convex k-gons in case k=12 (see our comment to A032279). (End)
%D A005516 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005516 Andrew Howroyd, <a href="/A005516/b005516.txt">Table of n, a(n) for n = 12..1000</a>
%H A005516 H. 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 A005516 W. D. Hoskins and Anne Penfold Street, <a href="http://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 A005516 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 A005516 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A005516 F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H A005516 V. 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 A005516 V. 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 A005516 A. P. Street, <a href="/A005513/a005513.pdf">Letter to N. J. A. Sloane, N.D.</a>
%H A005516 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%H A005516 <a href="/index/Rec#order_40">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-2,4,-4,12,-12,2,2,-12,24,-18,4,4,-6,15,-20,0,10,-4,10,0,-20,15,-6,4,4,-18,24,-12,2,2,-12,12,-4,4,-2,-4,4,-1).
%F A005516 Let s(n,k,d) = 1, if n==k (mod d), s(n,k,d) = 0, otherwise. Then a(n) = s(n,0,12)/6 + (n-6)*s(n,0,6)/72 + (n-4)*(n-8)*s(n,0,4)/384 + (n-3)*(n-6)*(n-9)*s(n,0,3)/1944 + (3840*C(n-1,11) + (n+1)*(n-2)*(n-4)*(n-6)*(n-8)*(n-10))/92160, if n is even; a(n) = (n-3)*(n-6)*(n-9)*s(n,0,3)/1944 + (3840*C(n-1,11) + (n-1)*(n-3)*(n-5)*(n-7)*(n-9)*(n-11))/92160, if n is odd. - _Vladimir Shevelev_, Apr 23 2011
%F A005516 From _Herbert Kociemba_, Nov 04 2016: (Start)
%F A005516 G.f.: 1/2*x^12*((1+x)/(1-x^2)^7 + 1/12*(1/(-1+x)^12 + 1/(-1+x^2)^6 + 2/(-1+x^3)^4 - 2/(-1+x^4)^3 + 2/(-1+x^6)^2 - 4/(-1+x^12))).
%F A005516 G.f.: k=12, x^k*((1/k)*(Sum_{d|k} phi(d)*(1 - x^d)^(-k/d)) + (1 + x)/(1 -x^2)^floor((k+2)/2))/2. (End)
%t A005516 k = 12; 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 A005516 k=12;CoefficientList[Series[x^k*(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[k/2+1])/2,{x,0,50}],x] (* _Herbert Kociemba_, Nov 04 2016 *)
%Y A005516 Column k=12 of A052307.
%K A005516 nonn
%O A005516 12,3
%A A005516 _N. J. A. Sloane_
%E A005516 Sequence extended and description corrected by _Christian G. Bower_