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A232642
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Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S, and duplicates are deleted as they occur.
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6
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1, 2, 4, 3, 6, 5, 10, 8, 7, 14, 12, 11, 22, 9, 18, 16, 15, 30, 13, 26, 24, 23, 46, 20, 19, 38, 17, 34, 32, 31, 62, 28, 27, 54, 25, 50, 48, 47, 94, 21, 42, 40, 39, 78, 36, 35, 70, 33, 66, 64, 63, 126, 29, 58, 56, 55, 110, 52, 51, 102, 49, 98, 96, 95, 190, 44
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OFFSET
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1,2
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COMMENTS
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Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,4), g(3) = (3,6,5,10), etc. Concatenating these gives A232642, a permutation of the positive integers. For n > 1, the number of numbers in g(n) is 2*F(n+1), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x + 2 if 2*x + 2 has not already occurred.
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LINKS
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EXAMPLE
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Each x begets x + 1 and 2*x + 2, but if either has already occurred it is deleted. Thus, 1 begets 2 and 4; then 2 begets 3 and 6, and 4 begets 5 and 10, so that g(3) = (3,6,5,10).
First 5 generations, also showing the places where duplicates were removed:
. 1: 1
. 2: 2 4
. 3: 3 6 5 10
. 4: _ 8 7 14 _ 12 11 22
. 5: _ __ 9 18 _ 16 15 30 _ __ 13 26 __ 24 23 46
These are the corresponding complete rows of triangle A082560:
. 1: 1
. 2: 2 4
. 3: 3 6 5 10
. 4: 4 8 7 14 6 12 11 22
. 5: 5 10 9 18 8 16 15 30 7 14 13 26 12 24 23 46
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MATHEMATICA
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z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 2]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]] (* A232642 *)
Table[Length[g1[n]], {n, 1, z}] (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]] (* A232643 *)
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PROG
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(Haskell)
import Data.List.Ordered (member); import Data.List (sort)
a232642 n k = a232642_tabf !! (n-1) !! (k-1)
a232642_row n = a232642_tabf !! (n-1)
a232642_tabf = f a082560_tabf [] where
f (xs:xss) zs = ys : f xss (sort (ys ++ zs)) where
ys = [v | v <- xs, not $ member v zs]
a232642_list = concat a232642_tabf
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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