cp's OEIS Frontend

A243611 Irregular triangular array of denominators of all rational numbers ordered as in Comments.

%I A243611 #5 Jun 11 2014 21:19:24
%S A243611 1,1,1,2,1,1,3,2,1,2,3,4,3,2,1,1,3,5,5,5,3,4,3,2,1,2,3,4,4,7,8,7,6,5,
%T A243611 5,5,3,4,3,2,1,1,3,5,5,5,7,8,9,7,11,11,9,7,4,7,8,7,6,5,5,5,3,4,3,2,1,
%U A243611 2,3,4,4,7,8,7,6,5,10,13,12,11,12,13,14
%N A243611 Irregular triangular array of denominators of all rational numbers ordered as in Comments.
%C A243611 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 A243611 Clark Kimberling, <a href="/A243611/b243611.txt">Table of n, a(n) for n = 1..3000</a>
%e A243611 First 6 rows of the array of all rationals:
%e A243611 0/1
%e A243611 -1/1 .. 1/1
%e A243611 -1/2 .. 2/1
%e A243611 -2/1 .. -1/3 .. 1/2 ... 3/1
%e A243611 -3/2 .. -2/3 .. -1/4 .. 2/3 ... 3/2 ... 4/1
%e A243611 -3/1 .. -4/3 .. -3/5 .. -2/5 .. -1/5 .. 1/3 . 3/4 . 5/3 . 5/2 . 5/1
%e A243611 The denominators, by rows:  1,1,1,2,1,1,3,2,1,2,3,4,3,2,1,1,3,5,5,3,4,3,2,1,...
%t A243611 z = 12; g[1] = {0}; f1[x_] := x + 1; f2[x_] := -1/(x + 1); h[1] = g[1];
%t A243611 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243611 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243611 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243611 u = Table[g[n], {n, 1, z}]
%t A243611 v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 2, z}]
%t A243611 Delete[Flatten[Denominator[u]], 6]  (* A243611 *)
%t A243611 Delete[Flatten[Numerator[u]], 6]    (* A243612 *)
%t A243611 Delete[Flatten[Denominator[v]], 2]  (* A243613 *)
%t A243611 Delete[Flatten[Numerator[v]], 2]    (* A243614 *)
%t A243611 ListPlot[g[20]]
%Y A243611 Cf. A243612, A243613, A243614, A226130, A000045.
%K A243611 nonn,easy,tabf,frac
%O A243611 1,4
%A A243611 _Clark Kimberling_, Jun 08 2014