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A032280
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Number of bracelets (turn over necklaces) of n beads of 2 colors, 7 of them black.
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3
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1, 1, 4, 8, 20, 38, 76, 133, 232, 375, 600, 912, 1368, 1980, 2829, 3936, 5412, 7293, 9724, 12760, 16588, 21287, 27092, 34112, 42640, 52819, 65008, 79392, 96405, 116280, 139536, 166464, 197676, 233529, 274740, 321741, 375364
(list; graph; refs; listen; history; internal format)
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OFFSET
| 7,3
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COMMENTS
| From Vladimir Shevelev, Apr 23 2011 (Start)
Also number of non-equivalent necklaces of 7 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=7 (see our comment to A032279).
(End)
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REFERENCES
| S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs ..., Z. Naturforsch., 52a (1997), 867-873.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
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LINKS
| C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
V. Shevelev,Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma)(Cf. Section 5)
Index entries for sequences related to bracelets
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FORMULA
| S. J. Cyvin et al. give a g.f.
"DIK[ 7 ]" (necklace, indistinct, unlabeled, 7 parts) transform of 1, 1, 1, 1...
From Vladimir Shevelev, 23 Apr 2011: (Start)
Put s(n,k,d)=1,if n==k(mod d); 0, otherwise. Then
a(n)=(3/7)*s(n,0,7)+(48*C(n-1,6)+7*(n-2)*(n-4)*(n-6))/672, if n is even; a(n)=(3/7)*s(n,0,7)+(48*C(n-1,6)+7*(n-1)*(n-3)*(n-5))/672, if n is odd.
(End)
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MATHEMATICA
| k = 7; 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 (russell(AT)post.harvard.edu), Sep 27 2004
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CROSSREFS
| Sequence in context: A097164 A133628 A097940 * A156303 A008136 A047196
Adjacent sequences: A032277 A032278 A032279 * A032281 A032282 A032283
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net)
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