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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 (list; graph; refs; listen; history; text; internal format)
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(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'

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

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Last modified December 19 08:25 EST 2018. Contains 318245 sequences. (Running on oeis4.)