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Irregular triangular array of denominators of all rational numbers ordered as in Comments.
5

%I #4 Jun 11 2014 21:19:55

%S 1,1,1,2,1,1,3,2,1,1,4,3,2,1,2,3,5,4,3,2,1,3,5,5,6,3,5,4,3,2,1,1,4,7,

%T 8,7,7,5,5,6,3,5,4,3,2,1,2,3,5,4,9,11,11,9,8,7,8,7,7,5,5,6,3,5,4,3,2,

%U 1,3,5,5,6,7,8,11,7,14,15,14,11,9,4,9,11

%N Irregular triangular array of denominators of all rational numbers ordered as in Comments.

%C Let W denote the array of all positive rational numbers defined at A243712. For the present array, put (row 1) = (1), (row 2) = (-1, 3), (row 3) = (-1/2,0,3), and (row 4) = (-1/3,1/2,4). Thereafter, (row n) consists of the following numbers in increasing order: (row n) of W together -1/x for each x in (row n-1) of W.

%H Clark Kimberling, <a href="/A243714/b243714.txt">Table of n, a(n) for n = 1..2000</a>

%e First 6 rows of the array of all positive rationals:

%e 1/1

%e -1/1 ... 2/1

%e -1/2 ... 0/1 ... 3/1

%e -1/3 ... 1/2 ... 4/1

%e -2/1 .... -1/4 ... 2/3 ... 3/2 ... 5/1

%e -3/2 ... -2/3 ... -1/5 ... 3/4 ... 5/3 ... 5/2 ... 6/1

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

%t 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]]]];

%t 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]

%t w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3];

%t u2 = Table[Drop[g[n], w[n]], {n, 1, z}];

%t u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4]

%t Denominator[u3] (* A243712 *)

%t Numerator[u3] (* A243713 *)

%t Denominator[u1] (* A243714 *)

%t Numerator[u1] (* A243715 *)

%Y Cf. A243712, A243713, A243715, A000930, A226130, A243613.

%K nonn,easy,tabf,frac

%O 1,4

%A _Clark Kimberling_, Jun 09 2014