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A243612 Irregular triangular array of numerators of all rational numbers ordered as in Comments. 5
0, -1, 1, -1, 2, -2, -1, 1, 3, -3, -2, -1, 2, 3, 4, -3, -4, -3, -2, -1, 1, 3, 5, 5, 5, -5, -5, -5, -3, -4, -3, -2, -1, 2, 3, 4, 4, 7, 8, 7, 6, -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, -7, -8, -9, -7, -11 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

LINKS

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

EXAMPLE

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 numerators, by rows:  0,-1, 1, -1, 2, -2, -1, 1, 3, -3, -2, -1, 2, 3, 4, -2, -4, -3, -2, -1, 1,3,5,5,5,...

MATHEMATICA

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]]

CROSSREFS

Cf. A243611, A243613, A243614, A226131, A000045.

Sequence in context: A126081 A268507 A272351 * A230351 A102481 A231201

Adjacent sequences:  A243609 A243610 A243611 * A243613 A243614 A243615

KEYWORD

easy,tabf,frac,sign

AUTHOR

Clark Kimberling, Jun 08 2014

STATUS

approved

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Last modified September 21 07:08 EDT 2019. Contains 327253 sequences. (Running on oeis4.)