0,3

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

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.

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

Another g.f.: (1+x^3)/((1-x)*(1-x^2)^2*(1-x^4)).

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

Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1]. - Michael Somos Feb 01 2007

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 ].

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.]

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)

(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

Cf. A006381, A006382.

Sequence in context: A043306 A131355 A092534 this_sequence A165272 A115264 A147617

Adjacent sequences: A005229 A005230 A005231 this_sequence A005233 A005234 A005235

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Sequence extended by Christian G. Bower (bowerc(AT)usa.net)