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

Author

Omar E. Pol, Jan 15 2014

Keywords

• nonn

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.

Examples

			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)
		

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).

Crossrefs

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

Cf. Companion of A235801.

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

Programs

Mathematica

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

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

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)