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A005516
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Number of n-bead bracelets (turn over necklaces) with 12 red beads.
(Formerly M4368)
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 12,3
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COMMENTS
| From Vladimir Shevelev, Apr 23 2011 (Start)
Also number of non-equivalent 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
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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
| 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
| 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(shevelev(AT)bgu.ac.il) Apr 23 2011.
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MATHEMATICA
| 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 (russell(AT)post.harvard.edu), Sep 27 2004
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CROSSREFS
| Sequence in context: A155333 A155296 A155463 * A152008 A002533 A111011
Adjacent sequences: A005513 A005514 A005515 * A005517 A005518 A005519
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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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