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

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, 34, 33, 31, 30, 29, 28, 26, 25, 11, 256, 96, 80, 72, 68, 66, 65, 63, 62, 61, 60, 58, 57, 56, 52, 50, 49, 23, 22, 21, 19, 512, 192, 160, 144, 136, 132, 130, 129, 127

Offset: 1

Author

Clark Kimberling, Jun 11 2014

Keywords

• nonn
• easy
• tabf

Comments

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.

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).

Examples

			First 7 rows of the array:
1
2
4
8 ... 3
16 .. 6 ... 5
32 .. 12 .. 10 .. 9 ... 7
64 .. 24 .. 20 .. 18 .. 17 .. 15 .. 14 .. 13
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1500

Crossrefs

Cf. A242364, A243735, A243736, A242448, A000045.

Programs

Mathematica

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}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)