OFFSET
1,1
COMMENTS
LINKS
FORMULA
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:
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. _ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _.
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. | | | | | | |_|_ _|_.
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. _ _|_ _|_ _|_ _|_ _|_ _ _|_ _|_ _|_.
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. _ _|_ _|_ _|_ _ _|_ _|_ _|_ _|_ _| .
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. 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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 15 2014
STATUS
approved