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Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
5

%I #7 Nov 09 2024 07:10:48

%S 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 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 20,13,12,15,12,13,10,8,7,8,7,4,13,17,23,27,29,29

%N Irregular triangular array of numerators 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="/A243731/b243731.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 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 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. A243730, A243712, A003269.

%K nonn,easy,tabf,frac,changed

%O 1,2

%A _Clark Kimberling_, Jun 09 2014