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A243574 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments. 5
1, 2, 3, 1, 3, 5, 2, 5, 8, 4, 3, 1, 4, 8, 7, 13, 7, 3, 5, 2, 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, 4, 8, 7, 13, 7, 3, 5, 2, 9, 18, 10, 19, 34, 18, 19, 9, 16, 31, 17, 31, 55, 29, 30, 14, 17, 29, 15, 14, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Decree that row 1 is (1) and row 2 is (1/2).  For n >=3, row n consists of numbers in increasing order generated  as follows:  1/(x + 1) for each x in row n-1 together with x + 1 for each x in row n-2.  It is easy to prove that row n consists of F(n) numbers, where F = A000045 (the Fibonacci numbers), and that every positive rational number occurs exactly once.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..6000

EXAMPLE

First 6 rows of the array of rationals:

1/1

1/2

2/3 ... 2/1

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

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

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

The denominators, by rows:  1,2,3,1,3,5,2,5,8,4,3,1,4,8,7,13,7,3,5,2,...

MATHEMATICA

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

Denominator[v]; (* A243574 *)

Numerator[v];   (* A242308 *)

CROSSREFS

Cf. A242308, A242359, A000045.

Sequence in context: A194760 A035517 A099471 * A121775 A127951 A208814

Adjacent sequences:  A243571 A243572 A243573 * A243575 A243576 A243577

KEYWORD

nonn,easy,tabf,frac

AUTHOR

Clark Kimberling, Jun 07 2014

STATUS

approved

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Last modified February 21 04:22 EST 2018. Contains 299389 sequences. (Running on oeis4.)