A243929 - OEIS

%I #6 Jun 19 2014 11:18:56

%S 1,-3,2,-2,-3,3,-1,-1,3,4,-3,0,1,5,5,6,-6,-6,-3,1,7,7,-12,-5,-6,-3,-1,

%T 2,5,9,8,-11,-15,-4,-12,-2,-3,1,4,3,4,7,9,11,9,15,-21,-10,-13,-21,-15,

%U -15,-7,-4,-6,-1,3,1,5,3,8,11,8,9,12,13,13,10,16,-20

%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 -3/x for each nonzero x in row n-1, where duplicates are deleted as they occur.  The number of numbers in row n is A243930(n).  Conjecture:  every rational number occurs exactly once in the array.

%H Clark Kimberling, <a href="/A243929/b243929.txt">Table of n, a(n) for n = 1..1000</a>

%e First 7 rows of the array of rationals:

%e 1/1

%e -3/1 .. 2/1

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

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

%e -3/4 .. 0/1 ... 1/2 ... 5/2 .. 5/1 .. 6/1

%e -6/1 .. -6/5 .. -3/5 .. 1/4 .. 7/2 .. 7/1

%e -12/1 . -5/1 .. -6/7 .. -3/7 . -1/5 . 2/5 . 5/4 . 9/2 . 8/1

%e The numerators, by rows:

%e   1,-3,2,-2,-3,3,-1,-1,3,4,-3,0,1,5,5,6,-6,6,-3,1,7,7,-12,-5,-6,-3,-1,2,5,9,8.

%t z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -3/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], 23]

%t Denominator[v] (* A243928 *)

%t Numerator[v]   (* A243929 *)

%Y Cf. A243928, A243930, A243925, A243712.

%K easy,tabf,frac,sign

%O 1,2

%A _Clark Kimberling_, Jun 15 2014