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A243730 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

%I A243730 #11 Nov 09 2024 07:01:37
%S A243730 1,1,1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,3,1,2,3,4,5,6,3,7,5,
%T A243730 5,1,2,3,4,5,6,3,7,5,5,8,7,8,7,1,2,3,4,5,6,3,7,5,5,8,7,8,7,9,9,11,11,
%U A243730 9,1,2,3,4,5,6,3,7,5,5,8,7,8,7,9,9,11
%N A243730 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
%C A243730 Suppose that m >= 3, and define sets h(n) of positive rational numbers as follows: h(n) = {n} for n = 1..m, and thereafter, h(n) = Union({x+1: x in h(n-1)}, {x/(x+1) : x in h(n-m)}), with the numbers in h(n) arranged in decreasing order. Every positive rational lies in exactly one of the sets h(n). For the present array, put m = 4 and (row n) = h(n); the number of numbers in h(n) is A003269(n-1). (For m = 3, see A243712.)
%H A243730 Clark Kimberling, <a href="/A243730/b243730.txt">Table of n, a(n) for n = 1..6000</a>
%e A243730 First 9 rows of the array:
%e A243730   1/1
%e A243730   2/1
%e A243730   3/1
%e A243730   4/1
%e A243730   5/1 .. 1/2
%e A243730   6/1 .. 3/2 .. 2/3
%e A243730   7/1 .. 5/2 .. 5/3 ... 3/4
%e A243730   8/1 .. 7/2 .. 8/3 ... 7/4 ... 4/5
%e A243730   9/1 .. 9/2 .. 11/3 .. 11/4 .. 9/5 .. 5/6 .. 1/3
%e A243730 The denominators, by rows:  1,1,1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,3,...
%t A243730 z = 27; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4};
%t A243730 g[n_] := Reverse[Union[1 + g[n - 1], g[n - 4]/(1 + g[n - 4])]]
%t A243730 Table[g[n], {n, 1, 12}]
%t A243730 v = Flatten[Table[g[n], {n, 1, z}]];
%t A243730 Denominator[v]; (* A243730 *)
%t A243730 Numerator[v];   (* A243731 *)
%Y A243730 Cf. A243731, A243712, A243732, A003269.
%K A243730 nonn,easy,tabf,frac
%O A243730 1,6
%A A243730 _Clark Kimberling_, Jun 09 2014