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%I M3311 #99 Feb 11 2021 09:01:51
%S 1,1,4,7,16,26,50,76,126,185,280,392,561,756,1032,1353,1782,2277,2920,
%T 3652,4576,5626,6916,8372,10133,12103,14448,17063,20128,23528,27474,
%U 31824,36822,42315,48564,55404,63133,71554,81004
%N Number of n-bead bracelets (turnover necklaces) of two colors with 6 red beads and n-6 black beads.
%C From _Vladimir Shevelev_, Apr 23 2011: (Start)
%C Also number of non-equivalent (turnover) necklaces of 6 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=6 (see our comment to A032279).
%C (End)
%C Also number of reverse invariant anonymous and neutral equivalence classes of preference profiles with 3 alternatives and (n-6) agents (IANC model). - _Alexander Karpov_, Apr 12 2018
%C Also the number of weighted cubic graphs with weight n derived from one of the 2 cubic graphs on 6 vertices (contributing to A321306). - _R. J. Mathar_, Nov 05 2018
%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 Vincenzo Librandi, <a href="/A005513/b005513.txt">Table of n, a(n) for n = 6..5000</a>
%H S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, <a href="http://zfn.mpdl.mpg.de/data/Reihe_A/52/ZNA-1997-52a-0867.pdf">Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem</a>, Z. Naturforsch., 52a (1997), 867-873.
%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="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 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 A. Karpov, <a href="https://publications.hse.ru/mirror/pubs/share/direct/217868605">An Informational Basis for Voting Rules</a>, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188
%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 S. J. Cyvin et al. (1997) give a g.f.
%F G.f.: (x^6/12)*(1/(1-x)^6+4/(1-x^2)^3+2/(1-x^3)^2+3/((1-x)^2*(1-x^2)^2)+2/(1-x^6)). - _Vladeta Jovovic_, Feb 28 2007
%F G.f.: x^6*(1-x+x^2+x^3+2*x^4+2*x^6+x^8-x^5) / ( (x^2-x+1)*(1+x+x^2)^2*(1+x)^3*(x-1)^6 ). - _R. J. Mathar_, Sep 18 2011
%F From _Vladimir Shevelev_, Apr 23 2011: (Start)
%F if n==0 mod 6, a(n)=(24*C(n-1,5)+3*(n+1)*(n-2)*(n-4)+16*n)/288;
%F if n==3 mod 6, a(n)=(24*C(n-1,5)+3*(n-1)*(n-3)*(n-5)+16*n-48)/288;
%F if n==2,4 mod 6, a(n)=(8*C(n-1,5)+(n+1)*(n-2)*(n-4))/96;
%F if n==1,5 mod 6, a(n)=(8*C(n-1,5)+(n-1)*(n-3)*(n-5))/96.
%F (End)
%p A005513 := proc(n) if n mod 6 = 0 then (24*binomial(n-1,5)+3*(n+1)*(n-2)*(n-4)+16*n)/288 elif n mod 6 = 3 then (24*binomial(n-1,5)+3*(n-1)*(n-3)*(n-5)+16*n-48)/288 elif n mod 6 = 2 or n mod 6 = 4 then (8*binomial(n-1,5)+(n+1)*(n-2)*(n-4))/96 elif n mod 6 = 1 or n mod 6 = 5 then (8*binomial(n-1,5)+(n-1)*(n-3)*(n-5))/96 fi: end: seq(A005513(n),n=6..44); # _Johannes W. Meijer_, Aug 11 2011
%t k = 6; 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=6; 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 *)
%t CoefficientList[Series[(1/12) (1/(1 - x)^6 + 4/(1 - x^2)^3 + 2/(1 - x^3)^2 + 3/((1 - x)^2 (1 - x^2)^2) + 2/(1 - x^6)), {x, 0, 43}], x] (* _Vincenzo Librandi_, Apr 24 2018 *)
%Y Column k=6 of A052307.
%K nonn,easy,nice
%O 6,3
%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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