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A329813
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a(n) = denominator(b(n)), where b(0) = b(1) = 1 and b(n) = n*b(n-1)/b(n-2) for n >= 1.
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1
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1, 1, 1, 1, 1, 1, 1, 2, 5, 5, 7, 14, 6, 15, 275, 11, 91, 637, 14, 200, 935, 187, 247, 12103, 245, 22, 47311, 4301, 247, 112385, 5733, 2772, 1372019, 11339, 38285, 398164, 2184, 86394, 23324323, 56695, 21793, 69977323, 77064, 49368, 434680565, 464899, 937099, 6018049778, 635778, 2128995, 93977938153
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OFFSET
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0,8
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COMMENTS
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This sequence is derived from a particular case of a general recurrence relation expressed by B(0) = x, B(1) = y and B(n) = n*B(n-1)/B(n-2), for n > 1 and {x,y} any pair of nonzero real numbers. Scatter plots of sequences of this kind exhibit a particular pattern that suggests the following conjecture:
lim_{n->infinity} B(6n+i)/(6n+i) = C_i and C_i != C_j for 0 < i < j < 7.
This means that B(n)/n approaches a cycle of six different constant values which depend on the particular chosen seed {x,y}. In this particular case the seed is {1,1} and the corresponding conjectured constant limits {C_1, C_2, C_3, C_4, C_5, C_6} are approximately {0.431, 0.615, 1.426, 2.319, 1.626, 0.701}. The corresponding constant limits for a generic seed {x,y} are respectively {C_1*y, C_2*y/x, C_3/x, C_4/y, C_5*x/y, C_6*x}. If x and y are not both positive then four of these constants are negative and two are positive.
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LINKS
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FORMULA
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a(n) = denominator(b(n)), where b(0) = b(1) = 1 and b(n) = n!/Product_{j=1..n-2} a(j), for n > 1.
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MATHEMATICA
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b[0]=1; b[1]=1;
b[n_]:=b[n]=n*b[n-1]/b[n-2]
(* Table[b[j], {j, 1, 2^10}]//ListPlot *)
Table[Denominator@b[j], {j, 0, 2^5}]
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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