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A226131
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Numerators of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)
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8
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1, 2, -1, 3, -1, 0, 4, -1, 1, 5, -1, 2, 3, -2, 6, -1, 3, 5, -3, 5, -2, 7, -1, 4, 7, -4, 8, -3, 7, -2, 1, 8, -1, 5, 9, -5, 11, -4, 11, -3, 2, 9, -2, 3, 4, -3, 9, -1, 6, 11, -6, 14, -5, 15, -4, 3, 14, -3, 5, 7, -5, 11, -2, 5, 8, -5, 7, -3, 10, -1, 7, 13, -7
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OFFSET
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1,2
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COMMENTS
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Let S be the set of numbers defined by these rules: 1 is in S, and if nonzero x is in S, then x + 1 and -1/x are in S. Then S is the set of all rational numbers, produced in generations as follows: g(1) = (1), g(2) = (2, -1), g(3) = (3, -1/2, 0), g(4) = (4, -1/3, 1/2), ... For n > 4, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements. Let S' denote the sequence formed by concatenating the generations.
A226130: Denominators of terms of S'
A226136: Positions of positive integers in S'
A226137: Positions of integers in S'
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LINKS
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EXAMPLE
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The denominators and numerators are read from the rationals in S':
1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...
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MATHEMATICA
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g[1] := {1}; z = 20; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]]&[DeleteCases[g[n - 1], 0]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; Flatten[Map[g, Range[7]]] (* ordered rationals *)
Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf A003410 *)
f = Flatten[Map[g, Range[z]]];
Take[Denominator[f], 100] (* A226130 *)
Take[Numerator[f], 100] (* A226131 *)
p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
p2 = Flatten[Table[Position[f, -n], {n, 0, z}]];
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CROSSREFS
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Cf. A226080 (rabbit ordering of positive rationals).
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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