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A242361 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments. 3
1, 1, 2, 2, 3, 3, 1, 3, 5, 3, 5, 2, 5, 8, 4, 5, 8, 4, 3, 1, 4, 8, 7, 13, 7, 4, 8, 7, 13, 7, 3, 5, 2, 7, 13, 7, 12, 21, 11, 11, 5, 7, 13, 7, 12, 21, 11, 11, 5, 5, 8, 4, 3, 1, 5, 11, 11, 21, 12, 9, 19, 18, 34, 19, 10, 18, 9, 5, 11, 11, 21, 12, 9, 19, 18, 34 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Let F = A000045 (the Fibonacci numbers).  To construct the array of positive rationals, decree that row 1 is (1) and row 2 is (2).   Thereafter, row n consists of the following numbers in increasing order: the F(n-2) numbers 1/x from numbers x > 1 in row n-1, together with the F(n-3) numbers 1 + 1/x from numbers x < 1 in row n - 1, together with the F(n - 2) numbers (2*x + 1)/ (x + 1) from numbers x in row n-2.  Row n consists of F(n) numbers ranging from 1/((n+1)/2) to n/2 if n is odd and from 2/(n-1) to (n+2)/2 if n is even.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..5000

EXAMPLE

First 6 rows of the array of rationals:

1/1

2/1

1/2 ... 3/2

2/3 ... 5/3 ... 3/1

1/3 ... 3/5 ... 4/3 ... 8/5 ... 5/2

2/5 ... 5/8 ... 3/4 ... 7/5 ... 13/8 .. 7/4 ... 8/3 ... 4/1

The denominators, by rows:  1,1,2,2,3,3,1,3,5,3,5,2,5,8,4,5,8,4,3,1,...

MATHEMATICA

z = 18; g[1] = {1}; f1[x_] := 1 + 1/x; 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}]; v = Flatten[u]; Length[v]

Denominator[v];  (* A242361 *)

Numerator[v];    (* A242363 *)

CROSSREFS

Cf. A242363, A242359, A243574, A000045.

Sequence in context: A128924 A239957 A230040 * A116464 A284532 A125585

Adjacent sequences:  A242358 A242359 A242360 * A242362 A242363 A242364

KEYWORD

nonn,easy,tabf,frac

AUTHOR

Clark Kimberling, Jun 08 2014

STATUS

approved

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Last modified April 18 22:02 EDT 2021. Contains 343090 sequences. (Running on oeis4.)