A235795 -id:A235795 - cp's OEIS frontend

A347270 Square array T(n,k) in which row n lists the 3x+1 sequence starting at n, read by antidiagonals upwards, with n >= 1 and k >= 0.

1, 2, 4, 3, 1, 2, 4, 10, 4, 1, 5, 2, 5, 2, 4, 6, 16, 1, 16, 1, 2, 7, 3, 8, 4, 8, 4, 1, 8, 22, 10, 4, 2, 4, 2, 4, 9, 4, 11, 5, 2, 1, 2, 1, 2, 10, 28, 2, 34, 16, 1, 4, 1, 4, 1, 11, 5, 14, 1, 17, 8, 4, 2, 4, 2, 4, 12, 34, 16, 7, 4, 52, 4, 2, 1, 2, 1, 2, 13, 6, 17, 8, 22

Offset: 1

Omar E. Pol, Aug 25 2021

This array gives all 3x+1 sequences.
The 3x+1 or Collatz problem is described in A006370.
Column k gives the image of n at the k-th step.
This infinite square array contains the irregular triangles A070165, A235795 and A347271.
For a piping diagram of the 3x+1 problem see A235800.

			The corner of the square array begins:
   1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
   2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...
   3,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...
   4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ...
   5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ...
   6, 3,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ...
   7,22,11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, 1, ...
   8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
   9,28,14, 7,22,11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, ...
  10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
  11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, ...
  12, 6, 3,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
  13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
  14, 7,22,11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, ...
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
J. C. Lagarias, The 3x+1 Problem: An Overview, arXiv:2111.02635 [math.NT], 2021.
Index entries for sequences related to 3x+1 (or Collatz) problem

Main diagonal gives A347272.
Parity of this sequence is A347283.
Largest value in row n gives A056959.
Number of nonpowers of 2 in row n gives A208981.
Some rows n are: A153727 (n=1), A033478 (n=3), A033479 (n=9), A033480 (n=15), A033481 (n=21), A008884 (n=27), A008880 (n=33), A008878 (n=39), A008883 (n=51), A008877 (n=57), A008874 (n=63), A258056 (n=75), A258098 (n=79), A008876 (n=81), A008879 (n=87), A008875 (n=95), A008873 (n=97), A008882 (n=99), A245671 (n=1729).
First four columns k are: A000027 (k=0), A006370 (k=1), A075884 (k=2), A076536 (k=3).
Cf. A006877, A014682, A057716, A070165, A078719, A135282, A235795, A235800, A235801, A347265, A347267, A347268, A347269, A347271, A347519.

T:= proc(n, k) option remember; `if`(k=0, n, (j->
      `if`(j::even, j/2, 3*j+1))(T(n, k-1)))
    end:
seq(seq(T(d-k, k), k=0..d-1), d=1..20);  # Alois P. Heinz, Aug 25 2021

T[n_, k_] := T[n, k] = If[k == 0, n, Function[j,
     If[EvenQ[j], j/2, 3*j + 1]][T[n, k - 1]]];
Table[Table[T[d - k, k], {k, 0, d - 1}], {d, 1, 20}] // Flatten (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

A235800 Length of n-th vertical line segment from left to right in a diagram of a two-dimensional version of the 3x+1 (or Collatz) problem.

Original entry on oeis.org

3, 1, 7, 2, 11, 3, 15, 4, 19, 5, 23, 6, 27, 7, 31, 8, 35, 9, 39, 10, 43, 11, 47, 12, 51, 13, 55, 14, 59, 15, 63, 16, 67, 17, 71, 18, 75, 19, 79, 20, 83, 21, 87, 22, 91, 23, 95, 24, 99, 25, 103, 26, 107, 27, 111, 28, 115, 29, 119, 30, 123, 31, 127, 32

Offset: 1

Omar E. Pol, Jan 15 2014

nonn

In the diagram every cycle is represented by a directed graph.
After (3x + 1) the next step is (3y + 1).
After (x/2) the next step is (y/2).
A235801(n) gives the length of n-th horizontal line segment in the same diagram.
Also A004767 and A000027 interleaved.

			The first part of the diagram in the first quadrant looks like this:
. . . . . . . . . . . . . . . . . . . . . . . .
.              _ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _.
.             |   |   |   |   |   |   |   |_|_.
.             |   |   |   |   |   |   |  _ _|_.
.             |   |   |   |   |   |   |_|_ _|_.
.             |   |   |   |   |   |  _ _|_ _|_.
.             |   |   |   |   |   |_|_ _|_ _|_.
.          _ _|_ _|_ _|_ _|_ _|_ _ _|_ _|_ _|_.
.         |   |   |   |   |   |_|_ _|_ _|_ _|_.
.         |   |   |   |   |  _ _|_ _|_ _|_ _|_.
.         |   |   |   |   |_|_ _|_ _|_ _|_ _|_.
.         |   |   |   |  _ _|_ _|_ _|_ _|_ _|_.
.         |   |   |   |_|_ _|_ _|_ _|_ _|_ _| .
.      _ _|_ _|_ _|_ _ _|_ _|_ _|_ _|_ _|     .
.     |   |   |   |_|_ _|_ _|_ _|_ _|         .
.     |   |   |  _ _|_ _|_ _|_ _|             .
.     |   |   |_|_ _|_ _|_ _|                 .
.     |   |  _ _|_ _|_ _|                     .
.     |   |_|_ _|_ _|                         .
.  _ _|_ _ _|_ _|                             .
. |   |_|_ _|                                 .
. |  _ _|                                     .
. |_|                                         .
. . . . . . . . . . . . . . . . . . . . . . . .
. 3,1,7,2,11...
From _Omar E. Pol_, Aug 25 2021: (Start)
The above diagram is the skeleton of a piping model of the 3x+1 or Collatz problem as shown below:
The model consists of pipes, 90-degree elbows and three types of pumps that propel the fluid through the pipes.
The corner of the infinite diagram looks like this:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
                            | |         | |         | |         | |  _ _ _  .
                            | |         | |         | |         | | |     |_.
                            | |         | |         | |         | | |  12  _.
                            | |         | |         | |        _| |_|_ v _| .
                            | |         | |         | |       |  ^  |_|_|_ _.
                            | |         | |         | |       |  11  _ _ _ _.
                            | |         | |         | |  _ _ _|_ _ _| | |   .
                 _ _ _ _ _ _|_|_ _ _ _ _|_|_ _ _ _ _|_|_|     |_ _ _ _|_|_ _.
                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _   10  _ _ _ _ _ _ _.
                | |         | |         | |        _| |_|_ v _|       | |   .
                | |         | |         | |       |  ^  |_| |_ _ _ _ _| |_ _.
                | |         | |         | |       |  9   _| |_ _ _ _ _| |_ _.
                | |         | |         | |  _ _ _|_ _ _| | |         | |   .
                | |         | |         | | |     |_ _ _ _| |_ _ _ _ _|_|_ _.
                | |         | |         | | |  8   _ _ _ _| |_ _ _ _ _ _ _ _.
                | |         | |        _| |_|_ v _|       | |         | |   .
                | |         | |       |  ^  |_| |_ _ _ _ _| |_ _ _ _ _|_|_ _.
                | |         | |       |  7   _| |_ _ _ _ _| |_ _ _ _ _ _ _ _.
                | |         | |  _ _ _|_ _ _| | |         | |         | |   .
                | |         | | |     |_ _ _ _| |_ _ _ _ _| |_ _ _ _ _| |   .
                | |         | | |  6   _ _ _ _| |_ _ _ _ _| |_ _ _ _ _ _|   .
                | |        _| |_|_ v _|       | |         | |               .
                | |       |  ^  |_|_|_ _ _ _ _|_|_ _ _ _ _| |               .
                | |       |  5   _ _ _ _ _ _ _ _ _ _ _ _ _ _|               .
                | |  _ _ _|_ _ _| | |         | |                           .
     _ _ _ _ _ _|_|_|     |_ _ _ _|_|_ _ _ _ _| |                           .
    |  _ _ _ _ _ _ _   4   _ _ _ _ _ _ _ _ _ _ _|                           .
    | |        _| |_|_ v _|       | |                                       .
    | |       |  ^  |_| |_ _ _ _ _| |                                       .
    | |       |  3   _| |_ _ _ _ _ _|                                       .
    | |  _ _ _|_ _ _| | |                                                   .
    | | |     |_ _ _ _| |                                                   .
    | | |  2   _ _ _ _ _|                                                   .
   _| |_|_ v _|                                                             .
  |  ^  |_| |                                                               .
  |  1   _ _|                                                               .
  |_ _ _|                                                                   .
.                                                                           .
On the main diagonal of the diagram appear the pumps labeled with the positive integers (A000027).
The pumps labeled with the numbers 2, 6, 8, 12, 14, 18, 20, 24, ... (the nonzero terms of A047238) receive the fluid from the EAST and propel it in a SOUTH direction. The fluid then passes through a 90-degree elbow and then heads WEST.
The pumps labeled with the numbers 4, 10, 16, 22, 28, 34, 40, ... (A016957) are of the type "TEE" as they have two side inlets and one outlet. These receive the fluid from the EAST and from the WEST and propel it in a SOUTH direction. The fluid then passes through a 90-degree elbow and then heads WEST.
The pumps labeled with the numbers 1, 3, 5, 7, 9, 11, 13, ... (A005408) receive the fluid from the EAST and propel it in the NORTH direction. The fluid then passes through a 90-degree elbow and then heads EAST.
Starting from the n-th pump we have that the fluid makes a path equivalent to the trayectory of the 3x+1 sequence starting at n. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A347270 (all 3x+1 sequences).
Cf. Companion of A235801.
Cf. A000027, A004767, A005408, A006370, A014682, A016957, A047238, A070165, A193356, A235795.

LinearRecurrence[{0,2,0,-1},{3,1,7,2},70] (* Harvey P. Dale, Sep 29 2016 *)

from _future_ import division
A235800_list = [4*(n//2) + 3 if n % 2 else n//2 for n in range(1,10**4)] # Chai Wah Wu, Sep 26 2016

a(n) = A006370(n) - A193356(n).
From Chai Wah Wu, Sep 26 2016: (Start)
a(n) = 2*a(n-2) - a(n-4) for n > 4.
G.f.: x*(x^2 + x + 3)/((x - 1)^2*(x + 1)^2). (End)

A375266 Irregular triangle read by rows in which row n lists the iterates of the A375265 map from n to 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 1, 5, 16, 8, 4, 2, 1, 6, 2, 1, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 8, 4, 2, 1, 9, 3, 1, 10, 5, 16, 8, 4, 2, 1, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 12, 4, 2, 1, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Offset: 1

Paolo Xausa, Aug 08 2024

By definition the trajectory ends when 1 is reached, so row 1 does not contain the cycle 1 -> 4 -> 2 -> 1.

See A375265 for links.

			Triangle begins:
   1;
   2,  1;
   3,  1;
   4,  2,  1;
   5, 16,  8,  4,  2,  1;
   6,  2,  1;
   7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10,  5, 16,  8,  4,  2,  1;
   8,  4,  2,  1;
   9,  3,  1;
  10,  5, 16,  8,  4,  2,  1;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..12824 (rows 1..300 of the triangle, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A375265, A375267 (# of iterations), A375268 (row sums), A375280 (row maxs).

Cf. A070165, A070168, A235795, A350279, A350548.

A375265[n_] := Which[Divisible[n, 3], n/3, Divisible[n, 2], n/2, True, 3*n + 1];
Array[NestWhileList[A375265, #, # > 1 &] &, 15]

A235801 Length of n-th horizontal line segment in a diagram of a two-dimensional version of the 3x+1 (or Collatz) problem.

Original entry on oeis.org

0, 1, 2, 3, 7, 5, 6, 7, 8, 9, 17, 11, 12, 13, 14, 15, 27, 17, 18, 19, 20, 21, 37, 23, 24, 25, 26, 27, 47, 29, 30, 31, 32, 33, 57, 35, 36, 37, 38, 39, 67, 41, 42, 43, 44, 45, 77, 47, 48, 49, 50, 51, 87, 53, 54, 55, 56, 57, 97, 59, 60, 61, 62, 63, 107, 65, 66

Offset: 0

Omar E. Pol, Jan 15 2014

nonn

In the diagram every cycle is represented by a directed graph.
After (3x + 1) the next step is (3y + 1).
After (x/2) the next step is (y/2).
A235800(n) gives the length of n-th vertical line segment, from left to right, in the same diagram.

			The first part of the diagram in the first quadrant:
. . . . . . . . . . . . . . . . . . . . . . . .
.              _ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _.
.             |   |   |   |   |   |   |   |_|_.
.             |   |   |   |   |   |   |  _ _|_.
.             |   |   |   |   |   |   |_|_ _|_.
.             |   |   |   |   |   |  _ _|_ _|_.
.             |   |   |   |   |   |_|_ _|_ _|_.
.          _ _|_ _|_ _|_ _|_ _|_ _ _|_ _|_ _|_.
.         |   |   |   |   |   |_|_ _|_ _|_ _|_.
.         |   |   |   |   |  _ _|_ _|_ _|_ _|_.
.         |   |   |   |   |_|_ _|_ _|_ _|_ _|_.
.         |   |   |   |  _ _|_ _|_ _|_ _|_ _|_.
.         |   |   |   |_|_ _|_ _|_ _|_ _|_ _| . 11
.      _ _|_ _|_ _|_ _ _|_ _|_ _|_ _|_ _|     . 17
.     |   |   |   |_|_ _|_ _|_ _|_ _|         .  9
.     |   |   |  _ _|_ _|_ _|_ _|             .  8
.     |   |   |_|_ _|_ _|_ _|                 .  7
.     |   |  _ _|_ _|_ _|                     .  6
.     |   |_|_ _|_ _|                         .  5
.  _ _|_ _ _|_ _|                             .  7
. |   |_|_ _|                                 .  3
. |  _ _|                                     .  2
. |_|                                         .  1
. . . . . . . . . . . . . . . . . . . . . . . .  0
.                                              a(n)
.
For an explanation of this diagram as the skeleton of a piping model see A235800. - _Omar E. Pol_, Dec 30 2021

Cf. A347270 (all 3x+1 sequences).
Companion of A235800.
Cf. A000027, A004767, A006370, A014682, A016957, A070165, A235795.

from _future_ import division
A235801_list = [n if n % 6 != 4 else 10*(n//6)+7 for n in range(10**4)] # Chai Wah Wu, Sep 26 2016

a(n) = 10*k - 3, if n is of the form (6*k-2), k>=1, otherwise a(n) = n.
From Chai Wah Wu, Sep 26 2016: (Start)
a(n) = 2*a(n-6) - a(n-12) for n > 11.
G.f.: x*(x^2 + 1)*(x^3 + 2*x^2 + 1)*(x^5 + x^4 + 2*x + 1)/(x^12 - 2*x^6 + 1). (End)

A347271 Irregular triangle T(n,k) read by rows in which row n lists the terms of the 3x+1 trajectory of n, but the row ends when a term is a power of 2 or when a term is less than n, with n >= 1 and k >= 0.

Original entry on oeis.org

1, 2, 3, 10, 5, 16, 4, 5, 16, 6, 3, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 8, 9, 28, 14, 7, 10, 5, 11, 34, 17, 52, 26, 13, 40, 20, 10, 12, 6, 13, 40, 20, 10, 14, 7, 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 16, 17, 52, 26, 13, 18, 9, 19, 58, 29, 88, 44, 22, 11

Offset: 1

Omar E. Pol, Aug 25 2021

Note that every row ends when it is easy to know the next missing terms because they are powers of 2 or the last term and the next missing terms form a row that it is already in the sequence.

For a square array with infinitely many terms in every row, see A347270, which is a supersequence that contains all 3x+1 sequences.

			Triangle begins:
   1;
   2;
   3,  10,   5,  16;
   4;
   5,  16;
   6,   3;
   7,  22,  11,  34,  17,  52,  26,  13,  40,  20,  10,   5;
   8;
   9,  28,  14,   7;
  10,   5;
  11,  34,  17,  52,  26,  13,  40,  20,  10;
  12,   6;
  13,  40,  20,  10;
  14,   7;
  15,  46,  23,  70,  35, 106,  53, 160,  80,  40,  20,  10;
  16;
  17,  52,  26,  13;
  18,   9;
  19,  58,  29,  88,  44,  22,  11;
...
For n = 3 the 3x+1 trajectory is 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... The fourth term is 16 which is a power of 2 so the third row of the triangle is [3, 10, 5, 16].
For n = 6 the 3x+1 trajectory is 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... The second term is 3 which is less than 6 so the 6th row of the triangle is [6, 3].

Index entries for sequences related to 3x+1 (or Collatz) problem

Subsequence of A070165, of A235795 and of A347270.

Cf. A000079, A006370, A014682, A056959, A235800, A263716.

A347653 Partial sums of the trajectory of all positive integers in the 3x+1 or Collatz problem, including the trajectory [1, 4, 2, 1] of 1.

Original entry on oeis.org

1, 5, 7, 8, 10, 11, 14, 24, 29, 45, 53, 57, 59, 60, 64, 66, 67, 72, 88, 96, 100, 102, 103, 109, 112, 122, 127, 143, 151, 155, 157, 158, 165, 187, 198, 232, 249, 301, 327, 340, 380, 400, 410, 415, 431, 439, 443, 445, 446, 454, 458, 460, 461, 470, 498, 512, 519, 541, 552, 586

Offset: 1

Omar E. Pol, Sep 09 2021

			The first two rows of A235795 are [1, 4, 2, 1]; [2, 1], so a(1)..a(6) are [1, 5, 7, 8, 10, 11].

Partial sums of A235795.

Cf. A006370, A235800, A347270 (all 3x+1 sequences).

A235795row[n_]:=If[n==1,{1,4,2,1},NestWhileList[If[OddQ[#],3#+1,#/2]&,n,#>1&]];
nrows=10;Accumulate[Flatten[Array[A235795row,nrows]]] (* Paolo Xausa, Jun 20 2022 *)

f(n) = if (n%2, 3*n+1, n/2); \\ A014682
row(n) = {my(list=List()); listput(list, n); until(n==1, n = f(n); listput(list, n)); Vec(list);} \\ A235795
lista(nn) = {my(s=0, list = List()); for (n=1, nn, my(v = row(n)); for (k=1, #v, s += v[k]; listput(list, s););); Vec(list);} \\ Michel Marcus, Sep 10 2021

A347652 Records in the trajectory of all positive integers in the 3x+1 or Collatz problem, including the trajectory [1, 4, 2, 1] of 1.

Original entry on oeis.org

1, 4, 10, 16, 22, 34, 52, 70, 106, 160, 214, 322, 484, 700, 790, 1186, 1780, 2158, 3238, 4858, 7288, 9232, 13120, 17224, 17494, 26242, 39364, 41524, 45682, 68524, 77092, 97576, 98962, 148444, 167002, 250504, 354292, 504466, 756700, 851290, 1276936, 1417174, 2125762

Offset: 1

Omar E. Pol, Sep 09 2021

nonn

Replacing the second term (4) with the first two primes (2, 3) we have 1, 2, 3, 10, 16, 22, ... the records in A070165.

			The first three rows of A235795 are [1, 4, 2, 1]; [2, 1]; [3, 10, 5, 16, 8, 4, 2, 1]. The records are [1, 4, 10, 16], the same as a(1)..a(4).

Index entries for sequences related to 3x+1 (or Collatz) problem

Records in A235795.

Cf. A006370, A070165, A235800, A347270 (all 3x+1 sequences).

f(n) = if (n%2, 3*n+1, n/2); \\ A014682
row(n) = {my(list=List()); listput(list, n); until(n==1, n = f(n); listput(list, n)); Vec(list);} \\ A235795
lista(nn) = {my(m=0, list = List()); for (n=1, nn, my(v = row(n)); for (k=1, #v, if (v[k]>m, m=v[k]; listput(list, m););)); Vec(list);} \\ Michel Marcus, Sep 10 2021

More terms from Michel Marcus, Sep 10 2021

A348136 Irregular triangle read by rows T(n,k) in which row n lists the first digit of every term of the trajectory of n in Collatz problem including the trajectory [1, 4, 2, 1] for n = 1.

Original entry on oeis.org

1, 4, 2, 1, 2, 1, 3, 1, 5, 1, 8, 4, 2, 1, 4, 2, 1, 5, 1, 8, 4, 2, 1, 6, 3, 1, 5, 1, 8, 4, 2, 1, 7, 2, 1, 3, 1, 5, 2, 1, 4, 2, 1, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 9, 2, 1, 7, 2, 1, 3, 1, 5, 2, 1, 4, 2, 1, 5, 1, 8, 4, 2, 1, 1, 5, 1, 8, 4, 2, 1, 1, 3, 1, 5, 2, 1, 4, 2, 1, 5, 1, 8, 4, 2, 1, 1, 6, 3, 1, 5, 1, 8, 4, 2, 1

Offset: 1

Omar E. Pol, Oct 02 2021

			Triangle begins:
1,4,2,1;
2,1;
3,1,5,1,8,4,2,1;
4,2,1;
5,1,8,4,2,1;
6,3,1,5,1,8,4,2,1;
7,2,1,3,1,5,2,1,4,2,1,5,1,8,4,2,1;
8,4,2,1;
9,2,1,7,2,1,3,1,5,2,1,4,2,1,5,1,8,4,2,1;
1,5,1,8,4,2,1;
1,3,1,5,2,1,4,2,1,5,1,8,4,2,1;
1,6,3,1,5,1,8,4,2,1;
1,4,2,1,5,1,8,4,2,1;
1,7,2,1,3,1,5,2,1,4,2,1,5,1,8,4,2,1;
...

First digit of every term of A235795.
First column gives A000030.
Cf. A006370, A014682, A070165, A235800, A347270 (all 3x+1 sequences).

Prepend[Map[First@ IntegerDigits[#] &, Array[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, #, # > 1 &] &, 11, 2], {2}], NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, 1, # > 1 &, {2, 1}]] // Flatten (* Michael De Vlieger, Oct 27 2021 *)

a(m) = A000030(A235795(m)), m >= 1.

T(n,k) = A000030(A235795(n,k)).

cp's OEIS Frontend

A347270 Square array T(n,k) in which row n lists the 3x+1 sequence starting at n, read by antidiagonals upwards, with n >= 1 and k >= 0.

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Maple

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A235800 Length of n-th vertical line segment from left to right in a diagram of a two-dimensional version of the 3x+1 (or Collatz) problem.

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A375266 Irregular triangle read by rows in which row n lists the iterates of the A375265 map from n to 1.

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Mathematica

A235801 Length of n-th horizontal line segment in a diagram of a two-dimensional version of the 3x+1 (or Collatz) problem.

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Python

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A347271 Irregular triangle T(n,k) read by rows in which row n lists the terms of the 3x+1 trajectory of n, but the row ends when a term is a power of 2 or when a term is less than n, with n >= 1 and k >= 0.

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A347653 Partial sums of the trajectory of all positive integers in the 3x+1 or Collatz problem, including the trajectory [1, 4, 2, 1] of 1.

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A347652 Records in the trajectory of all positive integers in the 3x+1 or Collatz problem, including the trajectory [1, 4, 2, 1] of 1.

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PARI

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A348136 Irregular triangle read by rows T(n,k) in which row n lists the first digit of every term of the trajectory of n in Collatz problem including the trajectory [1, 4, 2, 1] for n = 1.

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