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%I #4 Jun 11 2014 21:19:40
%S 0,-1,1,-1,2,-2,-1,1,3,-3,-2,-1,2,3,4,-3,-4,-3,-2,-1,1,3,5,5,5,-5,-5,
%T -5,-3,-4,-3,-2,-1,2,3,4,4,7,8,7,6,-4,-7,-8,-7,-6,-5,-5,-5,-3,-4,-3,
%U -2,-1,1,3,5,5,5,7,8,9,7,11,11,9,7,-7,-8,-9,-7,-11
%N Irregular triangular array of numerators of all rational numbers ordered as in Comments.
%C Let F = A000045 (the Fibonacci numbers). Row n of the array to be generated consists of F(n-1) nonnegative rationals together with F(n-1) negative rationals. The nonnegatives, for n >=3, are x + 1 from the F(n-2) nonnegative numbers x in row n-1, together with x/(x + 1) from the F(n-3) nonnegative numbers x in row n-2. The negatives in row n are the negative reciprocals of the positives in row n.
%H Clark Kimberling, <a href="/A243612/b243612.txt">Table of n, a(n) for n = 1..3000</a>
%e First 6 rows of the array of all rationals:
%e 0/1
%e -1/1 .. 1/1
%e -1/2 .. 2/1
%e -2/1 .. -1/3 .. 1/2 ... 3/1
%e -3/2 .. -2/3 .. -1/4 .. 2/3 ... 3/2 ... 4/1
%e -3/1 .. -4/3 .. -3/5 .. -2/5 .. -1/5 .. 1/3 . 3/4 . 5/3 . 5/2 . 5/1
%e The numerators, by rows: 0,-1, 1, -1, 2, -2, -1, 1, 3, -3, -2, -1, 2, 3, 4, -2, -4, -3, -2, -1, 1,3,5,5,5,...
%t z = 12; g[1] = {0}; f1[x_] := x + 1; f2[x_] := -1/(x + 1); h[1] = g[1];
%t 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]];
%t g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t u = Table[g[n], {n, 1, z}]
%t v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 2, z}]
%t Delete[Flatten[Denominator[u]], 6] (* A243611 *)
%t Delete[Flatten[Numerator[u]], 6] (* A243612 *)
%t Delete[Flatten[Denominator[v]], 2] (* A243613 *)
%t Delete[Flatten[Numerator[v]], 2] (* A243614 *)
%t ListPlot[g[20]]
%Y Cf. A243611, A243613, A243614, A226131, A000045.
%K easy,tabf,frac,sign
%O 1,5
%A _Clark Kimberling_, Jun 08 2014
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