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A248573 An irregular triangle giving the Collatz-Terras tree. 5
1, 2, 4, 8, 5, 16, 3, 10, 32, 6, 20, 21, 64, 12, 13, 40, 42, 128, 24, 26, 80, 84, 85, 256, 48, 17, 52, 53, 160, 168, 170, 512, 96, 11, 34, 104, 35, 106, 320, 336, 113, 340, 341, 1024 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Wolfdieter Lang, Oct 31 2014: (Start)

(old name corrected)

Irregular triangle CT(l, m) such that the first three rows l = 0, 1 and 2 are 1, 2, 4, respectively, and for l > 3 the row entries CT(l, m) are obtained from replacing the numbers of row l-1 by (2*x-1)/3, 2*x if they are 2 (mod 3) and by 2*x otherwise.

The modified Collatz (or Collatz-Terras) map sends a positive number x to x/2 if it is even and to (3*x+1)/2 if it is odd (see A060322). The present tree (without the complete tree originating at CT(2,1) = 1) can be considered as an incomplete binary tree, with nodes (vertices) of out-degree 2 if they are 2 (mod 3) and out-degree 1 otherwise. In the example below, the edges (branches) could be labeled L (left) or V (vertical).

The row length sequence is A060322(l+1), l>=0. (End)

The Collatz conjecture is true if and only if all odd numbers appear in this sequence.

This sequence is similar to A127824.

LINKS

Table of n, a(n) for n=0..43.

Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252.

Eric Weisstein's World of Mathematics, Collatz Problem.

EXAMPLE

The irregular triangle CT(l,m) begins:

l\m   1   2  3   4   5   6   7   8   9  10  11   12  13   14   15  16  17   18   19  20  21   22   23   24 ...

0:    1

1:    2

2:    4  here the 1, which would generate the complete tree again, is omitted

3:    8

4:    5  16

5:    3  10 32

6:    6  20 21  64

7:   12  13 40  42 128

8:   24  26 80  84  85 256

9:   48  17 52  53 160 168 170 512

10:  96  11 34 104  35 106 320 336 113 340 341 1024

11: 192   7 22  68  69 208  23  70 212 213 640  672  75  226  680 227 682 2048

12: 384  14 44  45 136 138 416  15  46 140 141  424 426 1280 1344 150 452  453 1360 151 454 1364 1365 4096

... reformatted, and extended -Wolfdieter Lang, Oct 31 2014

--------------------------------------------------------------------------------------------------------------

From Wolfdieter Lang, Oct 31 2014: (Start)

The Collatz-Terras tree starting with 4 looks like (numbers x == 2 (mod 3) are marked with a left bar, and the left branch ends then in (2*x-1)/3 and the vertical one in 2*x)

l=2:                                                                                        4

l=3:                                                                                       |8

l=4:                                                    |5                                 16

l=5:    3                                               10                                |32

l=6:    6                                              |20   21                            64

l=7:   12                     13                        40   42                          |128

l=8:   24                    |26                       |80   84            85             256

l=9:   48           |17       52              |53      160  168          |170            |512

l=10:  96     |11    34     |104        |35   106      320  336     |113  340      |341  1024

l=11: 192   7  22   |68  69  208   23|   70   212  213 640  672  75  226  680  227  682  2048

...

E.g., x = 7 = CT(11, 2) leads back to 4 via 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, and from there back to 2, 1.

(End)

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CROSSREFS

Cf. A127824, A060322, A088975.

Sequence in context: A036117 A307357 A116624 * A125733 A280426 A000689

Adjacent sequences:  A248570 A248571 A248572 * A248574 A248575 A248576

KEYWORD

nonn,tabf

AUTHOR

Nico Brown, Oct 08 2014

EXTENSIONS

Edited. New name (old corrected name as comment). - Wolfdieter Lang, Oct 31 2014

STATUS

approved

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Last modified February 27 03:50 EST 2020. Contains 332299 sequences. (Running on oeis4.)