A248573 An irregular triangle giving the Collatz-Terras tree.

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

Offset: 0

Author

Nico Brown, Oct 08 2014

Keywords

• nonn
• tabf

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.

Examples

			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
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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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Links

• Sebastian Karlsson, Rows l = 0..35, flattened
• Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252.
• Eric Weisstein's World of Mathematics, Collatz Problem.

Crossrefs

Cf. A127824, A060322, A088975.

Programs

Mathematica

Join[{{1}, {2}}, NestList[Flatten[Map[If[Mod[#, 3] == 2, {(2*#-1)/3, 2*#}, 2*#]&, #]]&, {4}, 10]] (* Paolo Xausa, Jan 25 2024 *)

PARI

rows(N) = my(r=List(),x); for(i=0, min(2, N), listput(r, x=[2^i])); for(n=3, N, my(w=List()); for(i=1, #x, my(q=2*x[i]); if(1==q%3, listput(w, (q-1)/3)); listput(w, q)); listput(r, x=Vec(w))); Vec(r); \\ Ruud H.G. van Tol, Jan 25 2024

Extensions

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