A243730 - OEIS

%I #11 Nov 09 2024 07:01:37

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

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

%C 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 Clark Kimberling, <a href="/A243730/b243730.txt">Table of n, a(n) for n = 1..6000</a>

%e First 9 rows of the array:

%e   1/1

%e   2/1

%e   3/1

%e   4/1

%e   5/1 .. 1/2

%e   6/1 .. 3/2 .. 2/3

%e   7/1 .. 5/2 .. 5/3 ... 3/4

%e   8/1 .. 7/2 .. 8/3 ... 7/4 ... 4/5

%e   9/1 .. 9/2 .. 11/3 .. 11/4 .. 9/5 .. 5/6 .. 1/3

%e 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 z = 27; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4};

%t g[n_] := Reverse[Union[1 + g[n - 1], g[n - 4]/(1 + g[n - 4])]]

%t Table[g[n], {n, 1, 12}]

%t v = Flatten[Table[g[n], {n, 1, z}]];

%t Denominator[v]; (* A243730 *)

%t Numerator[v];   (* A243731 *)

%Y Cf. A243731, A243712, A243732, A003269.

%K nonn,easy,tabf,frac

%O 1,6

%A _Clark Kimberling_, Jun 09 2014