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

0,2

Links

Table of n, a(n) for n=0..50.

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) ).

Example

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

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 *)

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.

Sequence in context: A277619 A001371 A339408 * A277631 A277633 A001037

Adjacent sequences: A277626 A277627 A277628 * A277630 A277631 A277632

Keyword

nonn,easy

Author

Herbert Kociemba, Oct 24 2016

Status

approved