A277633 Number of aperiodic necklaces (Lyndon words) with k<=8 black beads and n-k white beads.

1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 98, 180, 311, 546, 915, 1520, 2440, 3855, 5916, 8935, 13178, 19138, 27264, 38303, 52950, 72311, 97419, 129839, 171066, 223260, 288498, 369708, 469708, 592363, 741433, 921933, 1138761, 1398343, 1706956, 2072696, 2503513, 3009482, 3600515, 4289032, 5087253, 6010305, 7073122, 8293962

Offset: 0

Author

Herbert Kociemba, Oct 24 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1, 3, -2, -4, 1, 4, -1, -5, 2, 7, -1, -6, 0, 4, 0, -6, -1, 7, 2, -5, -1, 4, 1, -4, -2, 3, 1, -1).

Crossrefs

Cf. A001037 (k arbitrary), A008747 (k<=3), A277619 (k<=4), A277629 (k<=5), A277631 (k<=6).

The Mathematica section of A032168 gives the g.f. for k=m black beads and n-k white beads.

Programs

Mathematica

(* The g.f. for the number of aperiodic necklaces (Lyndon words) with k<=m black beads and n-k white beads. Here we have the case m=8 *)
gf[x_, m_]:=Sum[x^i/i Plus@@(MoebiusMu[#](1-x^#)^(-(i/#))&/@Divisors[i]), {i, 1, m}]+x+1

Formula

G.f.: 1 + x + x/(1-x) + 1/2*x^2*(1/(1-x)^2 - 1/(1-x^2)) + 1/3*x^3*(1/(1-x)^3 - 1/(1-x^3)) + 1/4*x^4*(1/(1-x)^4 - 1/(1-x^2)^2) + 1/5*x^5*(1/(1-x)^5 - 1/(1-x^5)) + 1/6*x^6*(1/(1-x)^6 - 1/(1-x^2)^3 - 1/(1-x^3)^2 + 1/(1-x^6)) + 1/7*x^7*(1/(1-x)^7 - 1/(1-x^7)) + 1/8*x^8*(1/(1-x)^8 - 1/(1-x^2)^4).