[FAQ] Sequence Vector

[mensa...@aol.com]

To navigate from one node on the Collatz tree to any other,
one follows a sequence of Rules defined by the Collatz
Conjecture. Although proper Collatz sequences always
move right and down on the tree, there is no mathematical
reason why one can't also move up and left (provided the
moves are legal) by inverting the standard Collatz Rules.

Thus, we have

Rule1: n/2
Rule2: n*3+1
InverseRule1: n*2
InverseRule2: (n-1)/3

A rule based data structure decribing a Collatz sequence
is called a

Sequence Vector.

By definition, a Sequence Vector always begins with an
odd number and always ends with at least 1 iteration of Rule1
(the terminating number may be even or odd).

For example, the sequence starting on 27 and ending on 31

27_82
41_124
62
31

is produced by the proper sequence

[Rule2,Rule1,Rule2,Rule1,Rule1]

We can abbreviate this by simply counting how many
consecutive iterations there are of the same rule.

[1*Rule2,1*Rule1,1*Rule2,2*Rule1]

Since a proper Collatz sequence cannot have more than 1
consecutive iteration of Rule2, we can further abbreviate
the sequence by assuming every block of Rule1 is seperated
by exactly 1 Rule2 and only showing the Rule1 blocks
(this assumption is why the Sequence Vector is defined
to start on an odd number and end with a Rule1).

[1*Rule1,2*Rule1]

Now, since every item shown is a block of Rule1, we can
further abbreviate by just showing the iteration count
of each block.

[1,2]

Thus, every proper Collatz sequence can be described by a
simple list of integers >0 (0 is not a valid Sequence
Vector element since the implied Rule2 to the left of each
element would result in two consecutive iterations of Rule2,
which is not permitted).

Sequence Vectors are not value-centric. They describe the
structure of the sequence without specifying values.
The sequence from 27 to 31 is a

Type [1,2] Sequence Vector

since the [1,2] pattern occurs infinitely many times on the
Collatz Tree.

3_10
5_16
8
4

11_34
17_52
26
13

19_58
29_88
44
22
.
.
.

Note that for the Type [1,2] Sequence Vector, the nth
occurence (for n:0,1,2,3...) starts at

n*8 + 3

and ends at

n*9 + 4

These are derived from any arbitrary Sequence Vector by
the Hailstone Function.

[Ernst]

That's a great post. Very clear.