

A226130


Denominators 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).


16



1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 3, 2, 1, 1, 5, 4, 3, 2, 2, 3, 1, 6, 5, 4, 3, 3, 5, 2, 5, 3, 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1, 1, 8, 7, 6, 5, 5, 9, 4, 11, 7, 3, 11, 8, 5, 2, 2, 9, 7, 5, 3, 3, 4, 1, 9, 8, 7, 6, 6, 11, 5, 14, 9, 4, 15, 11, 7, 3, 3
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OFFSET

1,5


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: A093394 A094363 A124832 * A137569 A266715 A089177
Adjacent sequences: A226127 A226128 A226129 * A226131 A226132 A226133


KEYWORD

nonn,frac


AUTHOR

Clark Kimberling, May 28 2013


STATUS

approved



