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A242365 Irregular triangular array of the positive integers ordered as in Comments.

%I A242365 #8 Jun 13 2014 05:04:43
%S A242365 1,2,4,8,3,16,6,5,32,12,10,9,7,64,24,20,18,17,15,14,13,128,48,40,36,
%T A242365 34,33,31,30,29,28,26,25,11,256,96,80,72,68,66,65,63,62,61,60,58,57,
%U A242365 56,52,50,49,23,22,21,19,512,192,160,144,136,132,130,129,127
%N A242365 Irregular triangular array of the positive integers ordered as in Comments.
%C A242365 As in A242364, let f1(x) = 2x, f2(x) = 1-x, f3(x) = 2-x, g(1) = (1), and g(n) = union(f1(g(n-1)), f2(g(n-1)),f3(g(n-1))) for n >1.  Let T be the array whose n-th row consists of the positive numbers in g(n) arranged in increasing order.  It is easy to prove that every positive integer occurs exactly once in T.
%C A242365 Conjectures:  (1)  |g(n)| = F(n-1) for n >=2, where F = A000045 (the Fibonacci numbers); (2) the number of even numbers in g(n) is F(n-2) and the number of odd numbers is F(n-3).
%H A242365 Clark Kimberling, <a href="/A242365/b242365.txt">Table of n, a(n) for n = 1..1500</a>
%e A242365 First 7 rows of the array:
%e A242365 1
%e A242365 2
%e A242365 4
%e A242365 8 ... 3
%e A242365 16 .. 6 ... 5
%e A242365 32 .. 12 .. 10 .. 9 ... 7
%e A242365 64 .. 24 .. 20 .. 18 .. 17 .. 15 .. 14 .. 13
%t A242365 z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
%t A242365 u1 = Flatten[u]  (* A242364 *)
%t A242365 v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
%t A242365 v1 = Flatten[v]  (* A242365 *)
%t A242365 w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
%t A242365 w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)
%Y A242365 Cf. A242364, A243735, A243736, A242448, A000045.
%K A242365 nonn,easy,tabf
%O A242365 1,2
%A A242365 _Clark Kimberling_, Jun 11 2014