%I #5 Jun 08 2014 09:18:19
%S 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,
%T 128,256,11,14,19,22,25,28,34,37,40,49,52,66,69,72,81,84,96,129,132,
%U 144,192,257,260,272,320,512,1024,15,23,26,29,35,38,41,44,50
%N Irregular triangular array generated as in Comments; contains every positive integer exactly once.
%C 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.
%H Clark Kimberling, <a href="/A243573/b243573.txt">Table of n, a(n) for n = 1..2500</a>
%e First 5 rows of the array:
%e 1
%e 2 .. 4
%e 3 .. 5 .. 8 .. 16
%e 6 .. 9 .. 12 . 17 . 20 . 32 . 64
%e 7 .. 10 . 13 . 18 . 21 . 24 . 33 . 36 . 48 . 65 . 68 . 80 . 128 . 256
%t z = 8; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 4 x; h[1] = g[1];
%t b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t u = Table[g[n], {n, 1, z}]; v = Flatten[u] (* A243573 *)
%Y Cf. A243571, A243572, A001631, A232563.
%K nonn,easy,tabf
%O 1,2
%A _Clark Kimberling_, Jun 07 2014