A226130 -id:A226130 - cp's OEIS frontend

A243611 Irregular triangular array of denominators of all rational numbers ordered as in Comments.

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 3, 4, 3, 2, 1, 1, 3, 5, 5, 5, 3, 4, 3, 2, 1, 2, 3, 4, 4, 7, 8, 7, 6, 5, 5, 5, 3, 4, 3, 2, 1, 1, 3, 5, 5, 5, 7, 8, 9, 7, 11, 11, 9, 7, 4, 7, 8, 7, 6, 5, 5, 5, 3, 4, 3, 2, 1, 2, 3, 4, 4, 7, 8, 7, 6, 5, 10, 13, 12, 11, 12, 13, 14

Offset: 1

Clark Kimberling, Jun 08 2014

Let F = A000045 (the Fibonacci numbers). Row n of the array to be generated consists of F(n-1) nonnegative rationals together with F(n-1) negative rationals. The nonnegatives, for n >=3, are x + 1 from the F(n-2) nonnegative numbers x in row n-1, together with x/(x + 1) from the F(n-3) nonnegative numbers x in row n-2. The negatives in row n are the negative reciprocals of the positives in row n.

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

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A243612, A243613, A243614, A226130, A000045.

z = 12; g[1] = {0}; f1[x_] := x + 1; f2[x_] := -1/(x + 1); 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}]
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 2, z}]
Delete[Flatten[Denominator[u]], 6]  (* A243611 *)
Delete[Flatten[Numerator[u]], 6]    (* A243612 *)
Delete[Flatten[Denominator[v]], 2]  (* A243613 *)
Delete[Flatten[Numerator[v]], 2]    (* A243614 *)
ListPlot[g[20]]

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 *)

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 *)

A233695 a(n) gives the position of -n in the sequence (or tree) S generated in order by these rules: 0 is in S; if x is in S then x + 1 is in S; if nonzero x is in S then 1/x is in S; if x is in S, then i*x is in S; where duplicates are deleted as they occur.

Original entry on oeis.org

10, 18, 30, 56, 109, 219, 450, 933, 1946, 4071, 8516, 17823, 37310, 78112, 163551, 342461, 717083, 1501509, 3144031, 6583341, 13784976

Offset: 1

Clark Kimberling, Dec 19 2013

It can be proved using the division algorithm for Gaussian integers that S is the set of Gaussian rational numbers: (b + c*i)/d, where b,c,d are integers and d is not 0.

Empirically, it appears that a(n) = A233694(n+2) + 7 for n > 2. It seems clear that positive integers appear for the first time at the start of a new level of the tree. If this is always the case, then the row starting with n will be followed by a row starting n+1, 1/n, ni, followed by a row starting n+2, 1/(n+1), (n+1)i, 1+1/n, n+1, i/(n+1), 1+ni, -i/n, -n. It may be possible to show that of these 9 values, only n+1 has ever appeared before. If so, then -n will always appear exactly 7 places after n + 2 in the sequence. - Jack W Grahl, Aug 10 2018

			The first 16 numbers generated are as follows:  0, 1, 2, i, 3, 1/2, 2 i, 1 + i, -i, -1, 4, 1/3, 3 i, 3/2, i/2, 1 + 2 i. -1 appears in the 10th place, so a(1) = 10.

Jack W Grahl, Haskell code to generate this sequence

Cf. A233694, A233696, A232559, A226130, A232723, A226080.

Off[Power::infy]; x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, 1/x, I*x} /. ComplexInfinity -> 0]]], {18}]; On[Power::infy]; t1 = Flatten[Position[x, _?(IntegerQ[#] && NonNegative[#] &)]]   (*A233694*)
t2 = Flatten[Position[x, _?(IntegerQ[#] && Negative[#] &)]] (* A233695 *)
t = Union[t1, t2]  (* A233696 *)
(* Peter J. C. Moses, Dec 21 2013 *)

More terms by Jack W Grahl, Aug 10 2018

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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A243611 Irregular triangular array of denominators of all 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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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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A233695 a(n) gives the position of -n in the sequence (or tree) S generated in order by these rules: 0 is in S; if x is in S then x + 1 is in S; if nonzero x is in S then 1/x is in S; if x is in S, then i*x is in S; where duplicates are deleted as they occur.

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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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