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

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.yu), 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 A115264 A147617 A072606

Adjacent sequences: A005229 A005230 A005231 this_sequence A005233 A005234 A005235

nonn,easy,nice

njas

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