A243715 -id:A243715 - cp's OEIS frontend

A226080 Denominators in the Fibonacci (or rabbit) ordering of the positive rational numbers.

1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 3, 1, 5, 4, 3, 4, 2, 5, 3, 1, 6, 5, 4, 5, 3, 7, 4, 2, 7, 5, 3, 5, 1, 7, 6, 5, 6, 4, 9, 5, 3, 10, 7, 4, 7, 2, 9, 7, 5, 7, 3, 8, 5, 1, 8, 7, 6, 7, 5, 11, 6, 4, 13, 9, 5, 9, 3, 13, 10, 7, 10, 4, 11, 7, 2, 11, 9, 7, 9, 5, 12, 7

Offset: 1

Clark Kimberling, May 25 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x+1 and 1/x are in S. Then S is the set of positive rational numbers, which arise in generations as follows: g(1) = (1/1), g(2) = (1+1) = (2), g(3) = (2+1, 1/2) = (3/1, 1/2), g(4) = (4/1, 1/3, 3/2), ... . Once g(n-1) = (g(1), ..., g(z)) is defined, g(n) is formed from the vector (g(1) + 1, 1/g(1), g(2) + 1, 1/g(2), ..., g(z) + 1, 1/g(z)) by deleting all elements that are in a previous generation. A226080 is the sequence of denominators formed by concatenating the generations g(1), g(2), g(3), ... . It is easy to prove the following:
(1) Every positive rational is in S.
(2) The number of terms in g(n) is the n-th Fibonacci number, F(n) = A000045(n).
(3) For n > 2, g(n) consists of F(n-2) numbers < 1 and F(n-1) numbers > 1, hence the name "rabbit ordering" since the n-th generation has F(n-2) reproducing pairs and F(n-1) non-reproducing pairs, as in the classical rabbit-reproduction introduction to Fibonacci numbers.
(4) The positions of integers in S are the Fibonacci numbers.
(5) The positions of 1/2, 3/2, 5/2, ..., are Lucas numbers (A000032).
(6) Continuing from (4) and (5), suppose that n > 0 and 0 < r < n, where gcd(n,r) = 1.  The positions in A226080 of the numbers congruent to r mod n comprise a row of the Wythoff array, W = A035513. The correspondence is sampled here:
   row 1 of W: positions of n+1 for n>=0,
   row 2 of W: positions of n+1/2,
   row 3 of W: positions of n+1/3,
   row 4 of W: positions of n+1/4,
   row 5 of W: positions of n+2/3,
   row 6 of W: positions of n+1/5,
   row 7 of W: positions of n+3/4.
(7) If the numbers <=1 in S are replaced by 1 and those >1 by 0, the resulting sequence is the infinite Fibonacci word A003849 (except for the 0-offset first term).
(8) The numbers <=1 in S occupy positions -1 + A001950, where A001950 is the upper Wythoff sequence; those > 1 occupy positions given by -1 + A000201, where A000201 is the lower Wythoff sequence.
(9) The rules (1 is in S, and if x is in S, then 1/x and 1/(x+1) are in S) also generate all the positive rationals.
A variant which extends this idea to an ordering of all rationals is described in A226130. - M. F. Hasler, Jun 03 2013
The updown and downup zigzag limits are (-1 + sqrt(5))/2 and (1 + sqrt(5))/2; see A020651. - Clark Kimberling, Nov 10 2013
From Clark Kimberling, Jun 19 2014: (Start)
Following is a guide to related trees and sequences; for example, the tree A226080 is represented by (1, x+1, 1/x), meaning that 1 is in S, and if x is in S, then x+1 and 1/x are in S (except for x = 0).
All the positive integers:
  A243571, A243572, A232559 (1, x+1, 2x)
  A232561, A242365, A243572 (1, x+1, 3x)
  A243573 (1, x+1, 4x)
All the integers:
  A243610 (1, 2x, 1-x)
  A232723, A242364
All the positive rationals:
  A226080, A226081, A242359, A242360 (1, x+1, 1/x)
  A243848, A243849, A243850 (1, x+1, 2/x)
  A243851, A243852, A243853 (1, x+1, 3/x)
  A243854, A243855, A243856 (1, x+1, 4/x)
  A243574, A242308 (1, 1/x, 1/(x+1))
  A241837, A243575 ({1,2,3}, x+4, 12/x)
  A242361, A242363 (1, 1 + 1/x, 1/x)
  A243613, A243614 (0, x+1, x/(x+1))
All the rationals:
  A243611, A243612 (0, x+1, -1/(x+1))
  A226130, A226131 (1, x+1, -1/x)
  A243712, A243713 ({1,2,3}, x+1, 1/(x+1))
  A243730, A243731 ({1,2,3,4}, x+1, 1/(x+1))
  A243732, A243733 ({1,2,3,4,5}, x+1, 1/(x+1))
  A243714, A243715
  A243925, A243926, A243927 (1, x+1, -2/x)
  A243928, A243929, A243930 (1, x+1, -3/x)
All the Gaussian integers:
  A243924 (1, x+1, i*x)
All the Gaussian rational numbers:
  A233694, A233695, A233696 (1, x+1, i*x, 1/x).
(End)

			The denominators are read from the rationals listed in "rabbit order":
1/1, 2/1, 3/1, 1/2, 4/1, 1/3, 3/2, 5/1, 1/4, 4/3, 5/2, 2/3, 6/1, ...

Clark Kimberling, Table of n, a(n) for n = 1..6000
Clark Kimberling, The infinite Fibonacci tree and other trees generated by rules, Proceedings of the 16th International Conference on Fibonacci Numbers and Their Applications, Fibonacci Quarterly 52 (2014), no. 5, pp. 136-149.
Index entries for fraction trees

Cf. A000045, A035513, A226081 (numerators), A226130, A226247, A020651.

z = 10; g[1] = {1}; g[2] = {2}; g[3] = {3, 1/2};
j[3] = Join[g[1], g[2], g[3]]; j[n_] := Join[j[n - 1], g[n]];
d[s_List, t_List] := Part[s, Sort[Flatten[Map[Position[s, #] &, Complement[s, t]]]]]; j[3] = Join[g[1], g[2], g[3]]; n = 3; While[n <= z, n++; g[n] = d[Riffle[g[n - 1] + 1, 1/g[n - 1]], g[n - 2]]];
Table[g[n], {n, 1, z}]; j[z] (* rabbit-ordered rationals *)
Denominator[j[z]]  (* A226080 *)
Numerator[j[z]]    (* A226081 *)
Flatten[NestList[(# /. x_ /; x > 1 -> Sequence[x, 1/x - 1]) + 1 &, {1}, 9]] (* rabbit-ordered rationals, Danny Marmer, Dec 07 2014 *)

A226080_vec(N=100)={my(T=[1],S=T,A=T); while(N>#A=concat(A, apply(denominator, T=select(t->!setsearch(S,t), concat(apply(t->[t+1,1/t],T))))), S=setunion(S,Set(T)));A} \\ M. F. Hasler, Nov 30 2018

(A226080(n)=denominator(RabbitOrderedRational(n))); ROR=List(1); RabbitOrderedRational(n)={if(n>#ROR, local(S=Set(ROR), i=#ROR*2\/(sqrt(5)+1), a(t)=setsearch(S,t)||S=setunion(S,[listput(ROR,t)])); until( type(ROR[i+=1])=="t_INT" && n<=#ROR, a(ROR[i]+1); a(1/ROR[i])));ROR[n]} \\ M. F. Hasler, Nov 30 2018

A243712 Irregular triangular array of denominators of all positive rational numbers ordered as in Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 09 2014

Decree that (row 1) = (1), (row 2) = (2), and (row 3) = (3). Thereafter, row n consists of the following numbers arranged in decreasing order: 1 + x for each x in (row n-1), together with x/(x + 1) for each x in row (n-3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n-1), for n >= 1.

			First 8 rows of the array of all positive rationals:
1/1
2/1
3/1
4/1 .. 1/2
5/1 .. 3/2 .. 2/3
6/1 .. 5/2 .. 5/3 ... 3/4
7/1 .. 7/2 .. 8/3 ... 7/4 ... 4/5 .. 1/3
8/1 .. 9/2 .. 11/3 .. 11/4 .. 9/5 .. 4/3 .. 5/6 .. 3/5 .. 2/5
The denominators, by rows: 1,1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,3,1,2,3,4,5,3,6,5,5,...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A243713, A243714, A243715, A000930, A243613.

z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -1/x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, z}]; u1 = Delete[Flatten[u], 10]
w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3];
u2 = Table[Drop[g[n], w[n]], {n, 1, z}];
u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4]
Denominator[u3]  (* A243712 *)
Numerator[u3]    (* A243713 *)
Denominator[u1]  (* A243714 *)
Numerator[u1]    (* A243715 *)

A243713 Irregular triangular array of numerators of all positive rational numbers ordered as in Comments.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 3, 2, 6, 5, 5, 3, 7, 7, 8, 7, 4, 1, 8, 9, 11, 11, 9, 4, 5, 3, 2, 9, 11, 14, 15, 14, 7, 11, 8, 7, 6, 5, 5, 3, 10, 13, 17, 19, 19, 10, 17, 13, 12, 13, 12, 13, 10, 7, 7, 8, 7, 4, 1, 11, 15, 20, 23, 24, 13, 23, 18, 17, 20, 19, 21, 17, 15, 16

Offset: 1

Clark Kimberling, Jun 09 2014

Decree that (row 1) = (1), (row 2) = (3), and (row 3) = (3). Thereafter, row n consists of the following numbers arranged in decreasing order: 1+x for each x in (row n-1), together with x/(x + 1) for each x in row (n-3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n-1), for n >= 1.

			First 8 rows of the array of all positive rationals:
1/1
2/1
3/1
4/1 ... 1/2
5/1 ... 3/2 ... 2/3
6/1 ... 5/2 ... 5/3 ... 3/4
7/1 ... 7/2 ... 8/3 ... 7/4 ... 4/5 ... 1/3
8/1 ... 9/2 ... 11/3 .. 11/4 .. 9/5 ... 4/3 ... 5/6 ... 3/5 ... 2/5
The numerators, by rows:  1,2,3,4,1,5,3,2,6,5,5,3,7,7,8,7,4,1,8,9,11,11,9,4,5,3,2,...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A243712, A243714, A243715, A000930, A243613.

z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -1/x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, z}]; u1 = Delete[Flatten[u], 10]
w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3];
u2 = Table[Drop[g[n], w[n]], {n, 1, z}];
u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4]
Denominator[u3]  (* A243712 *)
Numerator[u3]    (* A243713 *)
Denominator[u1]  (* A243714 *)
Numerator[u1]    (* A243715 *)

A243714 Irregular triangular array of denominators of all rational numbers ordered as in Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 09 2014

Let W denote the array of all positive rational numbers defined at A243712. For the present array, put (row 1) = (1), (row 2) = (-1, 3), (row 3) = (-1/2,0,3), and (row 4) = (-1/3,1/2,4). Thereafter, (row n) consists of the following numbers in increasing order: (row n) of W together -1/x for each x in (row n-1) of W.

			First 6 rows of the array of all positive rationals:
1/1
-1/1 ... 2/1
-1/2 ... 0/1 ... 3/1
-1/3 ... 1/2 ... 4/1
-2/1 .... -1/4 ... 2/3 ... 3/2 ... 5/1
-3/2 ... -2/3 ... -1/5 ... 3/4 ... 5/3 ... 5/2 ... 6/1
The denominators, by rows:  1,1,1,2,1,1,3,2,1,1,4,3,2,1,2,3,5,4,3,2,1,...

Clark Kimberling, Table of n, a(n) for n = 1..2000

Cf. A243712, A243713, A243715, A000930, A226130, A243613.

z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -1/x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, z}]; u1 = Delete[Flatten[u], 10]
w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3];
u2 = Table[Drop[g[n], w[n]], {n, 1, z}];
u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4]
Denominator[u3]  (* A243712 *)
Numerator[u3]    (* A243713 *)
Denominator[u1]  (* A243714 *)
Numerator[u1]    (* A243715 *)

cp's OEIS Frontend

A226080 Denominators in the Fibonacci (or rabbit) ordering of the positive rational numbers.

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A243712 Irregular triangular array of denominators of all positive rational numbers ordered as in Comments.

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A243713 Irregular triangular array of numerators of all positive rational numbers ordered as in Comments.

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A243714 Irregular triangular array of denominators of all rational numbers ordered as in Comments.

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