|
|
A242363
|
|
Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
|
|
3
|
|
|
1, 2, 1, 3, 2, 5, 3, 1, 3, 4, 8, 5, 2, 5, 3, 7, 13, 7, 8, 4, 1, 3, 4, 8, 5, 5, 11, 11, 21, 12, 7, 13, 7, 2, 5, 3, 7, 13, 7, 8, 4, 9, 18, 10, 19, 34, 18, 19, 9, 12, 21, 11, 11, 5, 1, 3, 4, 8, 5, 5, 11, 11, 21, 12, 7, 13, 7, 6, 14, 15, 29, 17, 14, 30, 29, 55
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
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
|
|
|
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 numerators, by rows: 1,2,1,3,2,5,3,1,3,4,8,5,2,5,3,7,13,7,8,4,...
|
|
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]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,tabf,frac
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|