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A242365 Irregular triangular array of the positive integers ordered as in Comments. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A109588 A254788 A052331 * A119436 A277695 A317503

Adjacent sequences:  A242362 A242363 A242364 * A242366 A242367 A242368

KEYWORD

nonn,easy,tabf

AUTHOR

Clark Kimberling, Jun 11 2014

STATUS

approved

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Last modified April 19 10:38 EDT 2019. Contains 322255 sequences. (Running on oeis4.)