OFFSET
1,2
COMMENTS
a(n-6) is the number of aperiodic necklaces (Lyndon words) with 6 black beads and n-6 white beads.
REFERENCES
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.
LINKS
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
FORMULA
G.f.: x*(1+x+2*x^2+2*x^3+3*x^4+2*x^6+x^7)/((1-x)^2*(1-x^2)^2*(1-x^3)*(1-x^6)).
G.f.: (1/(1-x)^6-1/(1-x^2)^3-1/(1-x^3)^2+1/(1-x^6))/6. - Herbert Kociemba, Oct 23 2016
a(n) = T(n,6), array T as in A051168.
MAPLE
a:= n-> (Matrix([[42, 20, 9, 3, 1, 0$7, -1, -4, -9]]). Matrix(15, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -3, -1, 1, 4, -3, -3, 4, 1, -1, -3, 1, 2, -1][i] else 0 fi)^(n-5))[1, 1]: seq(a(n), n=1..50); # Alois P. Heinz, Aug 04 2008
MATHEMATICA
a[n_] := Sum[Binomial[(n+6)/d, 6/d]*MoebiusMu[d], {d, Divisors[GCD[6, n+6]]}]/(n+6); Array[a, 40] (* Jean-François Alcover, Feb 02 2015 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved