Glide Chain Records

  1. The 3x+1 Glide Tree
  2. 3x + 1
  3. 3x + 5

The 3x+1 Glide Tree

Records in 3x+1

This illustration above shows the 3x+1 glide tree for odd numbers up to 199. We can use it to identify the following record holders.

Seed Value Glide Chain Length
3 1
7 2
9 3
25 4
33 5
97 6
129 7

So far there each record holder produces a chain one longer than the previous one.

To be continued.

Records in 3x+5

The situation in 3x+5 isn't so simple. There is no longer a single tree but at least six separate ones. We could generate a record set for each tree or just look to the length regardless of which tree the chain is in. There is a second complication as some numbers converge on loops where all the members are higher than the starting value, so they don't really count as glides. I've decided to redefine glides as being sequences that stop when they encounter a number which is either smaller than the seed or a member of a loop. The loops are represented by a single value (Ln) in the diagrams.

The first few record holders among all the trees are

Seed Value Glide Chain Length Tree
3 1 19
13 2 19
21 3 19
77 4 19
87 5 19
133 6 19

All of which are in the L19 tree.

To be continued.