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A032279
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Number of bracelets (turn over necklaces) of n beads of 2 colors, 5 of them black.
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12
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1, 1, 3, 5, 10, 16, 26, 38, 57, 79, 111, 147, 196, 252, 324, 406, 507, 621, 759, 913, 1096, 1298, 1534, 1794, 2093, 2421, 2793, 3199, 3656, 4152, 4706, 5304, 5967, 6681, 7467, 8311, 9234, 10222, 11298, 12446, 13691, 15015, 16445
(list; graph; refs; listen; history; internal format)
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OFFSET
| 5,3
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COMMENTS
| Contribution by Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of 5 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=5. The full solution was given by H.Gupta (1979); I gave a short proof of Gupta's result and showed an equivalence of this problem and every of the following problems: enumerating the bracelets of n beads of 2 colors, k of them black, and enumerating the necklaces of k beads each of them painted by one of n colors.
a(n) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order n with five 1's in every row.
(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
| Index entries for sequences related to bracelets
C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
V.Shevelev, A problem of enumeration of two-color bracelets with several variations
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
| "DIK[ 5 ]" (necklace, indistinct, unlabeled, 5 parts) transform of 1, 1, 1, 1...
G.f.: (1-x+2*x^3-x^5+x^6)/((1-x)^2*(1-x^2)^2*(1-x^5)).
Put s(n,k,d)=1, if n==k(mod d), 0, otherwise. Then
a(n)=0.4*s(n,0,5)+(n-1)*(n-3)*((n-2)*(n-4)+15)/240, if n is odd; a(n)=0.4*s(n,0,5)+(n-2)*(n-4)*((n-1)*(n-3)+15)/240, if n is even - Vladimir Shevelev, Apr 23 2011.
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MATHEMATICA
| k = 5; 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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PROG
| (PARI) a(n) =(n^4 -10*n^3 +50*n^2 -(110+30*(1-n%2))*n +(144-75*(n%2)+96*(1&(1<<(n%5)))))/240 - Washington Bomfim (webonfim(AT)bol.com.br), Jul 17 2008
(PARI) a(n) = round((n^4 -10*n^3 +50*n^2 -(110+30*(1-n%2))*n)/240 +3/5) - Washington Bomfim (webonfim(AT)bol.com.br), Jul 17 2008
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CROSSREFS
| Cf. A008805
Sequence in context: A006168 A037246 A184800 * A070558 A070559 A000990
Adjacent sequences: A032276 A032277 A032278 * A032280 A032281 A032282
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), N. J. A. Sloane (njas(AT)research.att.com).
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