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A248573
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An irregular triangle giving the Collatz-Terras tree.
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8
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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
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OFFSET
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0,2
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COMMENTS
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(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.
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LINKS
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EXAMPLE
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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
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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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MATHEMATICA
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Join[{{1}, {2}}, NestList[Flatten[Map[If[Mod[#, 3] == 2, {(2*#-1)/3, 2*#}, 2*#]&, #]]&, {4}, 10]] (* Paolo Xausa, Jan 25 2024 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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Edited. New name (old corrected name as comment). - Wolfdieter Lang, Oct 31 2014
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STATUS
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approved
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