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A277633
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Number of aperiodic necklaces (Lyndon words) with k<=8 black beads and n-k white beads.
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0
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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).
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MATHEMATICA
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(* 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
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CROSSREFS
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The Mathematica section of A032168 gives the g.f. for k=m black beads and n-k white beads.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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