cp's OEIS Frontend

A243731 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

%I A243731 #7 Nov 09 2024 07:10:48
%S A243731 1,2,3,4,5,1,6,3,2,7,5,5,3,8,7,8,7,4,9,9,11,11,9,5,1,10,11,14,15,14,
%T A243731 11,4,6,3,2,11,13,17,19,19,17,7,13,8,7,7,5,5,3,12,15,20,23,24,23,10,
%U A243731 20,13,12,15,12,13,10,8,7,8,7,4,13,17,23,27,29,29
%N A243731 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
%C A243731 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 A243731 Clark Kimberling, <a href="/A243731/b243731.txt">Table of n, a(n) for n = 1..6000</a>
%e A243731 First 9 rows of the array:
%e A243731   1/1
%e A243731   2/1
%e A243731   3/1
%e A243731   4/1
%e A243731   5/1 .. 1/2
%e A243731   6/1 .. 3/2 .. 2/3
%e A243731   7/1 .. 5/2 .. 5/3 ... 3/4
%e A243731   8/1 .. 7/2 .. 8/3 ... 7/4 ... 4/5
%e A243731   9/1 .. 9/2 .. 11/3 .. 11/4 .. 9/5 .. 5/6 .. 1/3
%e A243731 The numerators, by rows:  1,2,3,4,5,1,6,3,2,7,5,5,3,8,7,8,7,4,9,9,11,11,9,5,1...
%t A243731 z = 27; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4};
%t A243731 g[n_] := Reverse[Union[1 + g[n - 1], g[n - 4]/(1 + g[n - 4])]]
%t A243731 Table[g[n], {n, 1, 12}]
%t A243731 v = Flatten[Table[g[n], {n, 1, z}]];
%t A243731 Denominator[v];  (* A243730 *)
%t A243731 Numerator[v];  (* A243731 *)
%Y A243731 Cf. A243730, A243712, A003269.
%K A243731 nonn,easy,tabf,frac
%O A243731 1,2
%A A243731 _Clark Kimberling_, Jun 09 2014