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A005232
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G.f.: (1-x+x^2)/((1-x)^2*(1-x^2)*(1-x^4)).
(Formerly M2346)
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14
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1, 1, 3, 4, 8, 10, 16, 20, 29, 35, 47, 56, 72, 84, 104, 120, 145, 165, 195, 220, 256, 286, 328, 364, 413, 455, 511, 560, 624, 680, 752, 816, 897, 969, 1059, 1140, 1240, 1330, 1440, 1540, 1661, 1771, 1903, 2024, 2168, 2300, 2456, 2600, 2769, 2925, 3107, 3276
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of n-bead bracelets (turn over necklaces) with 4 red beads.
Also number of n X 2 binary matrices under row and column permutations and column complementations (if offset is 0).
Also Molien series for certain 4-D representation of dihedral group of order 8.
With offset 4, number of bracelets (turn over necklaces) of n-bead of 2 colors with 4 red beads. - Washington Bomfim (webonfim(AT)bol.com.br), Aug 27 2008
From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of 4 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=4 (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.
S. N. Ethier and S. E. Hodge, Identity-by-descent analysis of sibship configurations, Amer. J. Medical Genetics, 22 (1985), 263-272.
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.
M. Klemm, Selbstduale Codes ueber dem Ring der ganzen Zahlen modulo 4, Arch. Math. (Basel), 53 (1989), 201-207.
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
| T. D. Noe, Table of n, a(n) for n=0..1000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Index entries for sequences related to bracelets
Index entries for Molien series
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
| G.f.: (1+x^3)/((1-x)*(1-x^2)^2*(1-x^4)).
G.f.: (1/8)*(1/(1-x)^4+3/(1-x^2)^2+2/(1-x)^2/(1-x^2)+2/(1-x^4)) - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 05 2000
Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1]. - Michael Somos Feb 01 2007
a(2n+1) = A006918(2n+2)/2. a(2n)= (A006918(2n+1)+A008619(n))/2.
a(-6 - n) = -a(n) - Michael Somos Feb 05 2011
From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 22 2011: (Start)
if n==0 mod 4, then a(n)=n*(n^2-3*n+8)/48;
if n==1,3 mod 4, then a(n)=(n^2-1)*(n-3)/48;
if n==2 mod 4, then a(n)=(n-2)*(n^2-n+6)/48.
(End)
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EXAMPLE
| There are 8 4 X 2 matrices up to row and column permutations and column complementations:
[ 1 1 ] [ 1 0 ] [ 1 0 ] [ 0 1 ] [ 0 1 ] [ 0 1 ] [ 0 1 ] [ 0 0 ]
[ 1 1 ] [ 1 1 ] [ 1 0 ] [ 1 0 ] [ 1 0 ] [ 1 0 ] [ 0 1 ] [ 0 1 ]
[ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 0 ] [ 1 0 ] [ 1 0 ] [ 1 0 ]
[ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 0 ] [ 1 0 ] [ 1 1 ].
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MAPLE
| A005232:=-(-1-z-2*z**3+2*z**2+z**7-2*z**6+2*z**4)/(z**2+1)/(1+z)**2/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence apart from an initial 1.]
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MATHEMATICA
| k = 4; 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
CoefficientList[ Series[(1 - x + x^2)/((1 - x)^2(1 - x^2)(1 - x^4)), {x, 0, 51}], x] (from Robert G. Wilson v (rgwv(at)rgwv.com), Mar 29 2006)
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PROG
| (PARI) {a(n)=(n^3 +9*n^2 +(32-9*(n%2))*n +[48, 15, 36, 15][n%4+1])/48} /* Michael Somos Feb 01 2007 */
(PARI) {a(n)=local(s=1); if(n<-5, n=-6-n; s=-1); if(n<0, 0, s*polcoeff( (1-x+x^2)/ ((1-x)^2*(1-x^2)*(1-x^4)) +x*O(x^n), n))} /* Michael Somos Feb 01 2007 */
(PARI) a(n) = round((n^3 +9*n^2 +(32-9*(n%2))*n)/48 +0.6) - Washington Bomfim (webonfim(AT)bol.com.br), Jul 17 2008
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CROSSREFS
| Cf. A006381, A006382, A008805
Sequence in context: A043306 A131355 A092534 * A165272 A115264 A147617
Adjacent sequences: A005229 A005230 A005231 * A005233 A005234 A005235
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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 by Christian G. Bower (bowerc(AT)usa.net)
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