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A243926 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments. 4

%I #4 Jun 19 2014 11:18:28

%S 1,-2,2,-1,3,-2,0,4,-1,1,5,-6,-2,1,4,6,-5,-4,-3,-1,3,3,7,7,-10,-3,-4,

%T -6,-2,2,2,8,5,10,8,-7,-5,-4,-3,-1,1,5,7,5,13,7,13,9,-14,-14,-10,-6,

%U -10,-4,-6,-2,1,3,6,8,12,12,8,18,9,16,10,-13,-10,-8,-9

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

%C Decree that (row 1) = (1). For n >=2, row n consists of numbers in increasing order generated as follows: x+1 for each x in row n-1 together with -2/x for each nonzero x in row n-1, where duplicates are deleted as they occur. The number of numbers in row n is A243927(n). Conjecture: every rational number occurs exactly once in the array.

%H Clark Kimberling, <a href="/A243926/b243926.txt">Table of n, a(n) for n = 1..2500</a>

%e First 7 rows of the array of rationals:

%e 1/1

%e -2/1 ... 2/1

%e -1/1 ... 3/1

%e -2/3 ... 0/1 ... 4/1

%e -1/2 ... 1/3 ... 5/1

%e -6/1 ... -2/5 .. 1/2 ... 4/3 ... 6/1

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

%e The numerators, by rows: 1,-2,2,-1,3,-2,0,4,-1,1,5,-6,-2,1,4,6,-5,-4,-3,-1,3,3,7,7.

%t z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -2/x; h[1] = g[1];

%t b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];

%t h[n_] := h[n] = Union[h[n - 1], g[n - 1]];

%t g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]

%t u = Table[g[n], {n, 1, z}]

%t v = Delete[Flatten[u], 12]

%t Denominator[v] (* A243925 *)

%t Numerator[v] (* A243926 *)

%Y Cf. A243925, A243927.

%K easy,tabf,frac,sign

%O 1,2

%A _Clark Kimberling_, Jun 15 2014

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