I found another odd pattern by using the 3x + 1 function. I took each odd number, and I would then take the next odd number through the Collatz sequence (if I start with 7, I put it through the function to get 22, then 11). I then take that number and find the remainder of it and 4. The remainder of (11,4) is 3, so the function will have an ordered pair of (7,3).  The values are shown below. Sorry about the poor quality.

This is just a string of 1’s and 3’s. Like the pattern in the last blog, this has some predictability, but there is often a stray number that breaks up the order, in this case a 1 or 3. It is not impossible, though, to predict the next number at any point in the sequence.

The base sequence is 1,1,n,3,3,1,n,3. The n terms in the sequence are the ones that vary. The other terms continue to repeat every eight times. The n values, too, follow some predictability. If I “zoom out” on the pattern and only look at each n value, the pattern will look exactly like the base sequence.

The sequence of n terms here is 1,1,m,3,3,1,m,3.  The n terms that are not m terms will continue to repeat every 8 n terms (or every 32 regular terms). The m terms will just make the exact same base sequence when I zoom out, like the n terms before them.

This pattern of zooming out, only to find the base sequence again, goes on forever. There is absolutely no end to this; for every base sequence’s variation, there is a larger, identical sequence governing it.

This hierarchy of identical sequences isn’t limited to zooming out. I can find this identical base sequence when I zoom in and use even numbers as inputs, or even fractions. However, the base sequence for even numbers is shifted off. Although the sequence is still identical to the other sequences, the “starting point” is off the chart and appears to be where the input is -5. This is strange. If a mathematician uses this anomaly later in a proof, I found it first.

 

This pattern is closely related to the pattern I found last post, but the two patterns are not the same, as far as I know. This is another pattern that I could not find on the Internet. That makes me feel really special. I am going to name this sequence “The Moron Sequence”.