A226247 Let S be the set of numbers defined by these rules: 0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S. (See Comments.)

1, 1, 1, 1, 1, 2, 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

Offset: 1

Author

Clark Kimberling, Jun 01 2013

Keywords

• nonn
• easy
• tabf

Comments

Let S be the set of numbers defined by these rules: 0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S. Then S is the set of all rational numbers, produced in generations as follows:

g(1) = (0), g(2) = (1), g(3) = (2, -1), g(4) = (3, -1/2), g(5) = (4, -1/3, 1/2), ... For n > 2, 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.

A226247: Denominators of terms of S''

A226248: Numerators of terms of S''

A226249: Positions of nonnegative numbers in S''

A226250: Positions of positive numbers in S''

A closely related sequence S' (for which the rules of generation are shorter but the resulting sequence is slightly less natural) is discussed at A226130. For both S' and S'', the number of numbers in g(n) is given by A097333.

Examples

			The denominators and numerators are read from S'':
  0/1, 1/1, 2/1, -1/1, 3, -1/2, 4/1, -1/3, 1/2, 5, -1/4, 2/3, 3/2, -2, ...
Table begins:
  n |
  --+-----------------------------------------------
  1 | 1;
  2 | 1;
  3 | 1, 1;
  4 | 1, 2;
  5 | 1, 3, 2;
  6 | 1, 4, 3, 2, 1;
  7 | 1, 5, 4, 3, 2, 2, 3;
  8 | 1, 6, 5, 4, 3, 3, 5, 2, 5, 3;
  9 | 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1;
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Peter Kagey, Illustration of first seven generations.

Crossrefs

Cf. A226080 (rabbit ordering of positive rationals), A226130.

Cf. A226248, A226249, A226250.

Programs

Mathematica

z = 12; g[1] := {0}; g[2] := {1}; g[n_] :=  g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]] &[g[n - 1]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; f = Flatten[Map[g, Range[z]]]; Take[Denominator[f], 100]  (*A226247*)
t = Take[Numerator[f], 100]  (*A226248*)
s[n_] := If[t[[n]] > 0, 1, 0]; u = Table[s[n], {n, 1, Length[t]}]
Flatten[Position[u, 1]] (*A226249*)
p = Flatten[Position[u, 0]] (*A226250*) (* Peter J. C. Moses, May 30 2013 *)

Python

from fractions import Fraction
from itertools import count, islice
def agen():
    rats = [Fraction(0, 1)]
    seen = {Fraction(0, 1)}
    for n in count(1):
        yield from [r.denominator for r in rats]
        newrats = []
        for r in rats:
            f = 1+r
            if f not in seen:
                newrats.append(1+r)
                seen.add(f)
            if r != 0:
                g = -1/r
                if g not in seen:
                    newrats.append(-1/r)
                    seen.add(g)
        rats = newrats
print(list(islice(agen(), 84))) # Michael S. Branicky, Jan 17 2022