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

1,2

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).

Links

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A347669 A352812 A052331 * A119436 A277695 A353709

Adjacent sequences: A242362 A242363 A242364 * A242366 A242367 A242368

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Jun 11 2014

Status

approved