A243571 Irregular triangular array generated as in Comments; contains every positive integer exactly once.

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 13, 14, 17, 18, 20, 24, 32, 15, 19, 21, 22, 25, 26, 28, 33, 34, 36, 40, 48, 64, 23, 27, 29, 30, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 65, 66, 68, 72, 80, 96, 128, 31, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 67, 69

Offset

1,2

Comments

Decree that row 1 is (1) and row 2 is (2). For n >= 3, row n consists of numbers in increasing order generated as follows: 2*x for each x in row n-1 together with 1+2*x for each x in row n-2. It is easy to prove that row n consists of F(n) numbers, where F = A000045 (the Fibonacci numbers), and that every positive integer occurs exactly once. Row n has F(n-1) even numbers and F(n-2) odd numbers.

The least and greatest numbers in row n are A083329(n-1) and 2^(n-1), for n >= 1.

Links

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

Index entries for sequences that are permutations of the natural numbers

Example

First 6 rows of the array:
  1
  2
  3 ... 4
  5 ... 6 ... 8
  7 ... 9 ... 10 .. 12 .. 16
  11 .. 13 .. 14 .. 17 .. 18 .. 20 .. 24 .. 32

Mathematica

z = 10; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 2 x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[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, z}]; v = Flatten[u] (* A243571 *)

Crossrefs

Cf. A232559, A243572, A243573, A000045.

Cf. A052955 for the first element in each row.

Sequence in context: A362178 A358122 A334111 * A215366 A333483 A334433

Adjacent sequences: A243568 A243569 A243570 * A243572 A243573 A243574

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Jun 07 2014

Status

approved