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A243712 Irregular triangular array of denominators of all positive rational numbers ordered as in Comments. 12
1, 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 3, 1, 2, 3, 4, 5, 3, 6, 5, 5, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 8, 9, 11, 11, 9, 4, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 8, 9, 11, 11, 9, 4, 9, 11, 14, 15, 14, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Decree that (row 1) = (1), (row 2) = (2), and (row 3) = (3). Thereafter, row n consists of the following numbers arranged in decreasing order: 1 + x for each x in (row n-1), together with x/(x + 1) for each x in row (n-3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n-1), for n >= 1.
LINKS
EXAMPLE
First 8 rows of the array of all positive rationals:
1/1
2/1
3/1
4/1 .. 1/2
5/1 .. 3/2 .. 2/3
6/1 .. 5/2 .. 5/3 ... 3/4
7/1 .. 7/2 .. 8/3 ... 7/4 ... 4/5 .. 1/3
8/1 .. 9/2 .. 11/3 .. 11/4 .. 9/5 .. 4/3 .. 5/6 .. 3/5 .. 2/5
The denominators, by rows: 1,1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,3,1,2,3,4,5,3,6,5,5,...
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A133994 A066041 A194965 * A256553 A194896 A212721
KEYWORD
nonn,easy,tabf,frac
AUTHOR
Clark Kimberling, Jun 09 2014
STATUS
approved

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Last modified April 22 02:54 EDT 2024. Contains 371887 sequences. (Running on oeis4.)