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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

Cf. A232559, A232639, A232643, A000045.

Cf. A128588 (row lengths), A033484 (right edges), A257956 (row sums), A082560.

Sequence in context: A227113 A034701 A091857 * A180625 A132340 A132666

Adjacent sequences:  A232639 A232640 A232641 * A232643 A232644 A232645

KEYWORD

nonn,easy,tabf

AUTHOR

Clark Kimberling, Nov 28 2013

EXTENSIONS

Keyword tabf added, to bring out function g, by Reinhard Zumkeller, May 14 2015

STATUS

approved

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Last modified September 24 22:31 EDT 2017. Contains 292441 sequences.