6,3

The g.f. is Z(C_6,x)/x^6, the 6-variate cycle index polynomial for the cyclic group C_6, with substitution x[i]->1/(1-x^i), i=1,...,6. Therefore by Polya enumeration a(n+6) is the number of cyclically inequivalent 6-necklaces whose 6 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_6,x). Note the equivalence of this formulation with the one given in the `Name' line: start with a black 6-necklace (all 6 beads have labels 0). Insert after each of the 6 black beads k white ones if the label was k and then forget about the labels. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 15 2005.

C. G. Bower, Transforms (2)

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

Index entries for sequences related to necklaces

"CIK[ 6 ]" (necklace, indistinct, unlabeled, 6 parts) transform of 1, 1, 1, 1...

G.f.: [1-x+x^2+4x^3+2x^4+3x^6+x^7+x^8]/[(1-x)^6(1+x)^3(1+x+x^2)^2(1-x+x^2)] (conjectured). - R. Stephan, May 05 2004

G.f.:(x^6)*(1-x+x^2+4*x^3+2*x^4+3*x^6+x^7+x^8)/((1-x)^2*(1-x^2)^2*(1-x^3)*(1-x^6)) (proving the R. Stephan conjecture (with the correct offset) in a different version. W. Lang see above.)

k = 6; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] - Robert A. Russell (russell(AT)post.harvard.edu), Sep 27 2004

Cf. A004526, A007997, A008610, A008646.

Sequence in context: A155232 A023609 A055364 this_sequence A065568 A007825 A008256

Adjacent sequences: A032188 A032189 A032190 this_sequence A032192 A032193 A032194

nonn

Christian G. Bower (bowerc(AT)usa.net)