A277631 Number of aperiodic necklaces (Lyndon words) with k<=6 black beads and n-k white beads.

1, 2, 1, 2, 3, 6, 9, 18, 29, 51, 82, 135, 205, 315, 458, 662, 925, 1281, 1724, 2305, 3014, 3911, 4992, 6326, 7905, 9820, 12059, 14724, 17811, 21435, 25586, 30408, 35885, 42175, 49273, 57352, 66401, 76627, 88012, 100781, 114928, 130697, 148074, 167343, 188483, 211798, 237282, 265260, 295717, 329025, 365160

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-x^6 + 2*x^7 + 9*x^8 + 6*x^9 + 7*x^11 + 12*x^12 + 7*x^13 + 3*x^14 + 6*x^15 + 6*x^16 + x^17-3*x^18 + x^20)/( (-1+x)^6*(1+x)^3*(1-x+x^2)*(1+x+x^2)^2*(1+x+x^2+x^3+x^4) ).

Example

a(6)=9. The aperiodic necklaces are BWWWWW, BBWWWW, BWBWWW, BBBWWW, BBWBWW, BBWWBW, BBBBWW, BBBWBW and BBBBBW.

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=6 *)

Crossrefs

Cf. A001037 (k arbitrary), A008747 (k<=3), A277619 (k<=4), A277629 (k<=5). Mathematica section of A032168 gives g.f. for k=m black beads and n-k white beads.

Sequence in context: A001371 A339408 A277629 * A277633 A001037 A122086

Adjacent sequences: A277628 A277629 A277630 * A277632 A277633 A277634

Keyword

nonn,easy

Author

Herbert Kociemba, Oct 24 2016

Status

approved