## 2.10.16

### Collatz Exit Sequence Explained

As I already mentioned earlier tonight, a sequence I submitted to the OEIS on the 16th of September, 2016, a day after I wrote the following entry, has been rejected because it allegedly 'is very artificial".

The first few terms of this sequence are 5,27,21,107,85,427,341,1707,1365,6827, 5461

Here I present the recurrence relation, or formula, which I use to generate it.

a1 = 5

For i > 1, then:

ai = 5ai-1 + 2 if i is even
ai = ai-1 - (ai-2 + 1) if i is odd

Here's a simple program to generate the first i terms of the sequence in Javascript:

``````/* -------------------------------------------------
This content is released under the GNU License
http://www.gnu.org/copyleft/gpl.html
Author: Marina Ibrishimova
Version: 1.0
Purpose: Generate the sequence of Collatz exits and
Collatz traps
---------------------------------------------------- */
function generate_cet(n)
{
var a = new Array();
a.push(1);a.push(5);

for(i=2; i<=n; i++)
{
if(i%2 == 0){
a.push(5*(a[i-1]) + 2);
}
else{
a.push(a[i-1] - (a[i-2] +1));
}
}
return a;
}
``````

There were a few other recurrence relations from other members of the OEIS community to describe this sequence, but unfortunately these were destroyed when the sequence got rejected.

This sequence satisfies the definition of a mathematical sequence, and it has a clearly defined recurrence relation. The term "artificial" in relation to the term "mathematical sequence" does not exist. (yet)

In other words, there is absolutely nothing artificial about it.  (yet)

Furthermore, this sequence is an important clue in the Collatz conjecture.

I already authored 5 other sequences in the OEIS that I talked about here and here.

### Largest Power of 2

Recently, I showed a formula for finding an odd integer n = 5k + 2 based on another odd integer k such that  3k + 1 = 2i where i is the largest power of 2 such that  2i < n. I called k a Collatz exit point and n a Collatz trap.