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
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
Programs
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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).