A226275 -id:A226275 - 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

A226274 Position of 1/n in the ordering of the rationals given by application of the map t -> (1+t,-1/t), cf. A226130.

Original entry on oeis.org

1, 9, 31, 100, 317, 1000, 3150, 9918, 31223, 98289, 309406, 973981, 3065996, 9651448, 30381786, 95638797, 301061279, 947710512, 2983297009, 9391117780, 29562290606, 93059106094, 292940670339, 922147653681, 2902827709802, 9137808548505, 28764898718296, 90548996937472

Offset: 1

M. F. Hasler, Jun 01 2013

nonn

An analog of the Fibonacci ordering of the positive rationals (cf. A226080) is the sequence of rationals produced from the initial vector [1] by appending iteratively the new rationals obtained by applying the map t-> (t+1, -1/t) to each term of the vector (cf. example).

It is seen that the unit fraction 1/n appears as the last term produced in the (3n-3)th iteration, therefore the indices a(n) equal every third terms in the partial sums of A226275 (= new terms produced during the respective iteration), cf. formula.

			Starting with [1], applying the map t->(1+t,-1/t) to the (most recently obtained) vector and discarding the numbers occurring earlier, one gets the sequence (grouped by "generation"): [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],...
The unit fractions 1/1, 1/2, 1/3, 1/4,... occur at positions 1, 9(=1+2+3+3), 31(=9+5+7+10), 100(=31+15+22+32), ...

Cf. A226271, A226275, A226080, A226081, A226130.

{print1([s=1]", ");U=Set(g=[1]); for(n=1,29,U=setunion(U,Set(g=select(f->!setsearch(U,f), concat(apply(t->[t+1,if(t,-1/t)],g))))); for(i=1,#g, numerator(g[i])==1&&print1(s+i/*",g[i],*/","));s+=#g)} /* illustrative purpose only */

a(n) = s(3n-3) where s(k) = Sum_{j=0..k} A226275(j).

O.g.f.: x(1 + 4*x - 7*x^2 + 4*x^3 - x^4)/((1 - x)(1 - 4*x + 3*x^2 - x^3)).

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.)

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