A226136 -id:A226136 - cp's OEIS frontend

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.)

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

Offset: 1

Clark Kimberling, May 28 2013

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'
The length of row n is given by A226275(n-1). - Peter Kagey, Jan 17 2022

			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, ...
Table begins:
  n |
  --+-----------------------------------------------
  1 | 1;
  2 | 1, 1;
  3 | 1, 2, 1;
  4 | 1, 3, 2;
  5 | 1, 4, 3, 2, 1;
  6 | 1, 5, 4, 3, 2, 2, 3;
  7 | 1, 6, 5, 4, 3, 3, 5, 2, 5, 3;
  8 | 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1;

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

Cf. A226080 (rabbit ordering of positive rationals).
Cf. A226130, A226131, A226136, A226137, A226275.
Cf. A226247 (analogous with "0 is in S").

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 *)

from fractions import Fraction
from itertools import count, islice
def agen():
    rats = [Fraction(1, 1)]
    seen = {Fraction(1, 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

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.)

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 28 2013

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'

			Rationals in S': 1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...

Clark Kimberling, Table of n, a(n) for n = 1..1000
Peter Kagey, Illustration of the first seven generations.
Index entries for fraction trees

Cf. A226080 (rabbit ordering of positive rationals), A226247.

Cf. A226130, A226136, A226137.

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 *)

A226137 Positions of the integers in the ordering 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.)

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 14, 15, 22, 32, 46, 47, 69, 101, 147, 148, 217, 318, 465, 466, 683, 1001, 1466, 1467, 2150, 3151, 4617, 4618, 6768, 9919, 14536, 14537, 21305, 31224, 45760, 45761, 67066, 98290, 144050, 144051, 211117, 309407, 453457, 453458

Offset: 1

Clark Kimberling, May 28 2013

nonn

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'

			S'= (1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...), with integers appearing in positions 1,2,3,4,6,7,...

Clark Kimberling, Table of n, a(n) for n = 1..48

Cf. A226080 (rabbit ordering of positive rationals).

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 *)

A226275 Number of new rationals produced at the n-th iteration by applying the map t -> {t+1, -1/t} to nonzero terms, starting with S[0] = {1}.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452, 9651449

Offset: 0

M. F. Hasler, Jun 01 2013

nonn

The sequence produced by repeatedly applying t->(1+t,-1/t), starting from {1} and discarding numbers produced earlier, might be called Fibonacci or rabbit ordering of the rationals, in analogy to that ordering of the positive rationals, with t -> (1+t,1/t).

			The terms produced as described above are (grouped by iteration, including the starting value 1 = iteration 0): [1], [2, -1], [3, -1/2, 0], [4, -1/3, 1/2], [5, -1/4, 2/3, 3/2, -2], [6, -1/5, 3/4, 5/3, -3/2, 5/2, -2/3],[7, -1/6, 4/5, 7/4, -4/3, 8/3, -3/5, 7/2, -2/5, 1/3],[8, -1/7, 5/6, 9/5, -5/4, 11/4, -4/7, 11/3, -3/8, 2/5, 9/2, -2/7, 3/5, 4/3, -3], ...

Cf. A226271, A226274, A226080, A226081, A226130.

Essentially (up to initial terms) the same as A003410, A058278, A097333 and, in particular, A226136.

o.g.f. = (1 + x + x^2 - x^3 - x^5)/(1 - x - x^3)

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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.)

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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.)

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A226137 Positions of the integers in the ordering 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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A226275 Number of new rationals produced at the n-th iteration by applying the map t -> {t+1, -1/t} to nonzero terms, starting with S[0] = {1}.

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