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A005514
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Number of n-bead bracelets (turn over necklaces) with 8 red beads.
(Formerly M3801)
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4
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1, 1, 5, 10, 29, 57, 126, 232, 440, 750, 1282, 2052, 3260, 4950, 7440, 10824, 15581, 21879, 30415, 41470, 56021, 74503, 98254, 127920, 165288, 211276, 268228, 337416, 421856, 523260, 645456, 790704, 963793, 1167645, 1408185
(list; graph; refs; listen; history; internal format)
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OFFSET
| 8,3
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COMMENTS
| From Vladimir Shevelev, Apr 23 2011 (Start)
Also number of non-equivalent necklaces of 8 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=8 (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.
W. D. Hoskins; Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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.
G.f.: x^8/16*(1/(1 - x)^8 + 4/(1 - x^8) + 5/(1 - x^2)^4 + 2/(1 - x^4)^2 + 4/(1 - x)^2/(1 - x^2)^3) = x^8*(2*x^10 - 3*x^9 + 7*x^8 - 6*x^7 + 7*x^6 - 2*x^5 + 2*x^4 - 2*x^3 + 5*x^2 - 3*x + 1)/(1 - x)^8/(1 + x)^4/(1 + x^2)^2/(1 + x^4). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 17 2002
Contribution by Vladimir Shevelev, Apr 23 2011: (Start)
Let s(n,k,d)=1, if n==k(mod d), 0, otherwise. Then
a(n)=((n+4)/32)*s(n,0,8)+((n-4)/32)*s(n,4,8)+(48*C(n-1,7)+(n+1)*(n-2)*(n-4)*(n-6))/768, if n is even; a(n)=(48*C(n-1,7)+(n-1)*(n-3)*(n-5)*(n-7))/768, if n odd.
(End)
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MATHEMATICA
| k = 8; 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: A105862 A093029 A105505 * A069921 A053818 A133629
Adjacent sequences: A005511 A005512 A005513 * A005515 A005516 A005517
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Sequence extended and description corrected by Christian G. Bower (bowerc(AT)usa.net)
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