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A005513
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Number of n-bead bracelets (turnover necklaces) of two colors with 6 red beads and n-6 black beads.
(Formerly M3311)
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15
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1, 1, 4, 7, 16, 26, 50, 76, 126, 185, 280, 392, 561, 756, 1032, 1353, 1782, 2277, 2920, 3652, 4576, 5626, 6916, 8372, 10133, 12103, 14448, 17063, 20128, 23528, 27474, 31824, 36822, 42315, 48564, 55404, 63133, 71554, 81004
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OFFSET
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6,3
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COMMENTS
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Also number of non-equivalent (turnover) necklaces of 6 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=6 (see our comment to A032279).
(End)
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
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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.
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LINKS
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S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
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FORMULA
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S. J. Cyvin et al. (1997) give a g.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
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
if n==0 mod 6, a(n)=(24*C(n-1,5)+3*(n+1)*(n-2)*(n-4)+16*n)/288;
if n==3 mod 6, a(n)=(24*C(n-1,5)+3*(n-1)*(n-3)*(n-5)+16*n-48)/288;
if n==2,4 mod 6, a(n)=(8*C(n-1,5)+(n+1)*(n-2)*(n-4))/96;
if n==1,5 mod 6, a(n)=(8*C(n-1,5)+(n-1)*(n-3)*(n-5))/96.
(End)
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MAPLE
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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
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MATHEMATICA
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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 *)
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 *)
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 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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