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%I #18 Apr 09 2023 18:15:13
%S 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,
%T 28,33,34,36,40,48,64,23,27,29,30,35,37,38,41,42,44,49,50,52,56,65,66,
%U 68,72,80,96,128,31,39,43,45,46,51,53,54,57,58,60,67,69
%N Irregular triangular array generated as in Comments; contains every positive integer exactly once.
%C 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.
%C The least and greatest numbers in row n are A083329(n-1) and 2^(n-1), for n >= 1.
%H Clark Kimberling, <a href="/A243571/b243571.txt">Table of n, a(n) for n = 1..1500</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e First 6 rows of the array:
%e 1
%e 2
%e 3 ... 4
%e 5 ... 6 ... 8
%e 7 ... 9 ... 10 .. 12 .. 16
%e 11 .. 13 .. 14 .. 17 .. 18 .. 20 .. 24 .. 32
%t z = 10; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 2 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] (* A243571 *)
%Y Cf. A232559, A243572, A243573, A000045.
%Y Cf. A052955 for the first element in each row.
%K nonn,easy,tabf
%O 1,2
%A _Clark Kimberling_, Jun 07 2014
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