%I A005232 M2346
%S A005232 1,1,3,4,8,10,16,20,29,35,47,56,72,84,104,120,145,165,195,220,256,286,
%T A005232 328,364,413,455,511,560,624,680,752,816,897,969,1059,1140,1240,1330,
%U A005232 1440,1540,1661,1771,1903,2024,2168,2300,2456,2600,2769,2925,3107,3276
%N A005232 G.f.: (1-x+x^2)/((1-x)^2*(1-x^2)*(1-x^4)).
%C A005232 Number of n-bead bracelets (turn over necklaces) with 4 red beads.
%C A005232 Also number of n X 2 binary matrices under row and column permutations 
               and column complementations (if offset is 0).
%C A005232 Also Molien series for certain 4-D representation of dihedral group of 
               order 8.
%C A005232 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
%D A005232 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005232 S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs 
               ..., Z. Naturforsch., 52a (1997), 867-873.
%D A005232 S. N. Ethier and S. E. Hodge, Identity-by-descent analysis of sibship 
               configurations, Amer. J. Medical Genetics, 22 (1985), 263-272.
%D A005232 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.
%D A005232 M. Klemm, Selbstduale Codes ueber dem Ring der ganzen Zahlen modulo 4, 
               Arch. Math. (Basel), 53 (1989), 201-207.
%H A005232 T. D. Noe, <a href="b005232.txt">Table of n, a(n) for n=0..1000</a>
%H A005232 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/
               ~njas/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, 
               Springer, Berlin, 2006.
%H A005232 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005232 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005232 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">
               Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A005232 <a href="Sindx_Br.html#bracelets">Index entries for sequences related 
               to bracelets</a>
%H A005232 <a href="Sindx_Mo.html#Molien">Index entries for Molien series</a>
%F A005232 Another g.f.: (1+x^3)/((1-x)*(1-x^2)^2*(1-x^4)).
%F A005232 Another 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
%F A005232 Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1]. - Michael 
               Somos Feb 01 2007
%e A005232 There are 8 4 X 2 matrices up to row and column permutations and column 
               complementations:
%e A005232 [ 1 1 ] [ 1 0 ] [ 1 0 ] [ 0 1 ] [ 0 1 ] [ 0 1 ] [ 0 1 ] [ 0 0 ]
%e A005232 [ 1 1 ] [ 1 1 ] [ 1 0 ] [ 1 0 ] [ 1 0 ] [ 1 0 ] [ 0 1 ] [ 0 1 ]
%e A005232 [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 0 ] [ 1 0 ] [ 1 0 ] [ 1 0 ]
%e A005232 [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 0 ] [ 1 0 ] [ 1 1 ].
%p A005232 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.]
%t A005232 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
%t A005232 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)
%o A005232 (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 */
%o A005232 (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 */
%o A005232 (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
%Y A005232 Cf. A006381, A006382.
%Y A005232 Sequence in context: A043306 A131355 A092534 this_sequence A165272 A115264 
               A147617
%Y A005232 Adjacent sequences: A005229 A005230 A005231 this_sequence A005233 A005234 
               A005235
%K A005232 nonn,easy,nice
%O A005232 0,3
%A A005232 N. J. A. Sloane (njas(AT)research.att.com).
%E A005232 Sequence extended by Christian G. Bower (bowerc(AT)usa.net)