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Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
3

%I #11 Nov 09 2024 18:11:05

%S 1,2,3,4,5,6,1,7,3,2,8,5,5,3,9,7,8,7,4,10,9,11,11,9,5,11,11,14,15,14,

%T 11,6,1,12,13,17,19,19,17,13,4,7,3,2,13,15,20,23,24,23,20,7,15,8,7,8,

%U 5,5,3,14,17,23,27,29,29,27,10,23,13,12,17,12,13

%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 = 5 and (row n) = h(n); the number of numbers in h(n) is A003520(n-1). (For m = 3, see A243712.)

%H Clark Kimberling, <a href="/A243733/b243733.txt">Table of n, a(n) for n = 1..999</a>

%e First 11 rows of the array:

%e 1/1

%e 2/1

%e 3/1

%e 4/1

%e 5/1

%e 6/1 ... 1/2

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

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

%e 9/1 ... 7/2 ... 8/3 ... 7/4 ... 4/5

%e 10/1 .. 9/2 ... 11/3 .. 11/4 .. 9/5 ... 5/6

%e 11/1 .. 11/2 .. 14/3 .. 15/4 .. 14/5 .. 11/6 .. 6/7 .. 1/3

%e The numerators, by rows: 1,2,3,4,5,6,1,7,3,2,8,5,5,3,9,7,8,7,4,10,9,...

%t z = 23; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4}; g[5] = {5};

%t g[n_] := Reverse[Union[1 + g[n - 1], g[n - 5]/(1 + g[n - 5])]]

%t Table[g[n], {n, 1, 9}]

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

%t v1 = Denominator[v]; (* A243732 *)

%t v2 = Numerator[v]; (* A243733 *)

%Y Cf. A243732, A243713, A243731, A003520.

%K nonn,easy,tabf,frac,changed

%O 1,2

%A _Clark Kimberling_, Jun 09 2014