A243573 Irregular triangular array generated as in Comments; contains every positive integer exactly once.

1, 2, 4, 3, 5, 8, 16, 6, 9, 12, 17, 20, 32, 64, 7, 10, 13, 18, 21, 24, 33, 36, 48, 65, 68, 80, 128, 256, 11, 14, 19, 22, 25, 28, 34, 37, 40, 49, 52, 66, 69, 72, 81, 84, 96, 129, 132, 144, 192, 257, 260, 272, 320, 512, 1024, 15, 23, 26, 29, 35, 38, 41, 44, 50

Offset

1,2

Comments

Decree that (row 1) = (1), (row 2) = (2, 4), (row 3) = (3,5,8,16), (row 4) = (6,9,12,17,20,32,64). Let r(n) = A001563(n+3), so that r(r) = r(n-1) + r(n-2) + r(n-3) + r(n-4) with r(1) =1, r(2) = 2, r(3) = 4, r(4) = 7. Row n of the array, for n >= 5, consists of the numbers, in increasing order, defined as follows: all 4*x from x in row n-1, together with all 1 + 4*x from x in row n-2, together with all 2 + 4*x from x in row n-3, together with all 3 + 4*x for x in row n-4. Thus, the number of numbers in row n is r(n), a tetranacci number. Every positive integer occurs exactly once in the array, so that the resulting sequence is a permutation of the positive integers.

Links

Clark Kimberling, Table of n, a(n) for n = 1..2500

Example

First 5 rows of the array:
1
2 .. 4
3 .. 5 .. 8 .. 16
6 .. 9 .. 12 . 17 . 20 . 32 . 64
7 .. 10 . 13 . 18 . 21 . 24 . 33 . 36 . 48 . 65 . 68 . 80 . 128 . 256

Mathematica

z = 8; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 4 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] (* A243573 *)

Crossrefs

Cf. A243571, A243572, A001631, A232563.

Sequence in context: A360434 A215898 A246162 * A132193 A185910 A306779

Adjacent sequences: A243570 A243571 A243572 * A243574 A243575 A243576

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Jun 07 2014

Status

approved