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A005515
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Number of n-bead bracelets (turn over necklaces) with 10 red beads.
(Formerly M4105)
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3
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1, 1, 6, 14, 47, 111, 280, 600, 1282, 2494, 4752, 8524, 14938, 25102, 41272, 65772, 102817, 156871, 235378, 346346, 502303, 716859, 1010256, 1404624, 1931540, 2625658, 3534776, 4711448, 6226148, 8156396, 10603704
(list; graph; refs; listen; history; internal format)
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OFFSET
| 10,3
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COMMENTS
| From Vladimir Shevelev, Apr 23 2011 (Start)
Also number of non-equivalent necklaces of 10 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=10 (see our comment to A032279).
(End)
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REFERENCES
| 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.
Index entries for sequences related to bracelets
V. Shevelev,Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma)(Cf. Section 5)
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FORMULA
| Vladimir Shevelev, Apr 23 2011: (Start)
Put s(n,k,d)=1, if n==k(mod d), s(n,k,d)=0, otherwise. Then a(n)=n*s(n,0,5)/25+((384*C(n-1,9)+(n+1)*(n-2)*(n-4)*(n-6)*(n-8))/7680, if n is even; a(n)=(n-5)*s(n,0,5)/25+((384*C(n-1,9)+(n-1)*(n-3)*(n-5)*(n-7)*(n-9))/7680, if n is odd.
(End)
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MATHEMATICA
| k = 10; 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: A015892 A093369 A130443 * A114705 A200187 A107301
Adjacent sequences: A005512 A005513 A005514 * A005516 A005517 A005518
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KEYWORD
| nonn
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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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