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.

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

1,1

Comments

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.

Links

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).

Formula

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)

Example

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)

Mathematica

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

Prog

(Python)
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

Crossrefs

Cf. A347270 (all 3x+1 sequences).

Cf. Companion of A235801.

Cf. A000027, A004767, A005408, A006370, A014682, A016957, A047238, A070165, A193356, A235795.

Sequence in context: A065287 A065263 A057114 * A065259 A065289 A065265

Adjacent sequences: A235797 A235798 A235799 * A235801 A235802 A235803

Keyword

nonn

Author

Omar E. Pol, Jan 15 2014

Status

approved