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A005516
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Number of n-bead bracelets (turnover necklaces) with 12 red beads.
(Formerly M4368)
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3
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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
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OFFSET
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12,3
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COMMENTS
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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)
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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).
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LINKS
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FORMULA
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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
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)*(Sum_{d|k} phi(d)*(1 - x^d)^(-k/d)) + (1 + x)/(1 -x^2)^floor((k+2)/2))/2. (End)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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