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

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

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).

A235800(n) gives the length of n-th vertical line segment, from left to right, in the same diagram.

Examples

			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
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

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

Companion of A235800.

Cf. A000027, A004767, A006370, A014682, A016957, A070165, A235795.

Programs

Python

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

Formula

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)