

A226131


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).


8



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

1,2


COMMENTS

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(n1) = (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'
A226131: Numerators of terms of S'
A226136: Positions of positive integers in S'
A226137: Positions of integers in S'


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for fraction trees


EXAMPLE

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, ...


MATHEMATICA

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}]];
Union[p1, p2] (* A226137 *) (* Peter J. C. Moses, May 26 2013 *)


CROSSREFS

Cf. A226080 (rabbit ordering of positive rationals).
Sequence in context: A186976 A160550 A172038 * A199056 A144966 A320000
Adjacent sequences: A226128 A226129 A226130 * A226132 A226133 A226134


KEYWORD

sign,frac


AUTHOR

Clark Kimberling, May 28 2013


STATUS

approved



