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

0,2

Links

Table of n, a(n) for n=0..47.

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).

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).

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

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.

Sequence in context: A339408 A277629 A277631 * A001037 A122086 A082594

Adjacent sequences: A277630 A277631 A277632 * A277634 A277635 A277636

Keyword

nonn,easy

Author

Herbert Kociemba, Oct 24 2016

Status

approved