A277629 Number of aperiodic necklaces (Lyndon words) with k<=5 black beads and n-k white beads.

1, 2, 1, 2, 3, 6, 9, 17, 26, 42, 62, 93, 130, 183, 246, 329, 428, 553, 698, 877, 1082, 1328, 1608, 1937, 2307, 2736, 3215, 3762, 4369, 5055, 5810, 6657, 7584, 8614, 9737, 10976, 12320, 13795, 15388, 17126, 18997, 21029, 23208, 25565, 28085, 30799, 33694, 36801, 40105, 43641, 47392

Offset: 0

Author

Herbert Kociemba, Oct 24 2016

Keywords

• nonn
• easy

Examples

			a(5)=6: The aperiodic necklaces are BWWWW, BBWWW, BWBWW, BBBWW, BBWBW, BBBBW.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1, 2, -1, -2, 0, 1, -1, 0, 2, 1, -2, -1, 1).

Crossrefs

Cf. A001037 (k arbitrary), A008747 (k<=3), A277619 (k<=4). Mathematica section of A032168 gives 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 is *)
gf[x_, m_]:=Sum[x^i/i Plus@@(MoebiusMu[#](1-x^#)^(-(i/#))&/@Divisors[i]), {i, 1, m}]+x+1
(* Here we have the case m=5 *)

Formula

G.f.: (-1 - x + 3*x^2 + 2*x^3 - 3*x^4 - 4*x^5 - 2*x^7 - 4*x^8 + 3*x^10 - 2*x^11 - 4*x^12 + x^14)/( (-1+x)^5*(1+x)^2*(1+x+x^2)*(1+x+x^2+x^3+x^4) ).