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 A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads. (Formerly M4368) 2
 1, 1, 7, 19, 72, 196, 561, 1368, 3260, 7105, 14938, 29624, 56822, 104468, 186616, 322786, 544802, 896259, 1444147, 2278640, 3532144, 5380034, 8070400, 11926928, 17393969, 25042836, 35638596, 50152013, 69855536 (list; graph; refs; listen; history; text; internal format)
 OFFSET 12,3 COMMENTS From Vladimir Shevelev, Apr 23 2011: (Start) Also number of non-equivalent (turnover) necklaces of 12 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=12 (see our comment to A032279). (End) REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999. W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15. W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy) 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.) A. P. Street, Letter to N. J. A. Sloane, N.D. FORMULA 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 From Herbert Kociemba, Nov 04 2016: (Start) 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))). G.f.: k=12, x^k*(1/k*Plus@@(EulerPhi[#]*(1-x^#)^(-(k/#))&/@Divisors[k]) + (1+x)/(1-x^2)^Floor[(k+2)/2])/2. (End) MATHEMATICA 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 *) 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 *) CROSSREFS Sequence in context: A155296 A155463 A318483 * A152008 A002533 A111011 Adjacent sequences:  A005513 A005514 A005515 * A005517 A005518 A005519 KEYWORD nonn AUTHOR EXTENSIONS Sequence extended and description corrected by Christian G. Bower STATUS approved

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Last modified July 17 16:38 EDT 2019. Contains 325107 sequences. (Running on oeis4.)