A232642 - OEIS

A232642

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.

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

1,2

COMMENTS

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.

Seen as triangle read by rows: A082560 with duplicates removed. - Reinhard Zumkeller, May 14 2015

LINKS

Clark Kimberling and Reinhard Zumkeller, Rows n = 1..17 of triangle, flattened, first 13 rows from Clark Kimberling

Index entries for sequences that are permutations of the natural numbers

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(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

-- Reinhard Zumkeller, May 14 2015