Curious About Symmetric Number Sequences

[hariz91]

TL;DR

I've been exploring symmetric sequences like 1,2,1 and 1,2,3,2,1. The middle number seems to produce a new value. I'm curious if this pattern has a name or if anyone has explored it before.

I’ve been exploring number patterns as a personal interest — particularly symmetric sequences like 1, 2, 1 or 1, 2, 3, 2, 1. I noticed that the middle value in these sequences sometimes seems to produce a new number when combined in a certain way. For example:

- 1, [2], 1 → gives 0
- 1, 2, [3], 2, 1 → gives 3
- 1, 2, 3, [4], 3, 2, 1 → gives 8
- 1, 2, 3, 4, [5], 4, 3, 2, 1 → gives 5
...
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1 → gives 63

I’m not sure if this pattern is already known or if it has a name. I’m just curious — has anyone seen something like this before? I’d appreciate any thoughts or references.

 

[jedishrfu]

How are you getting those results?

 

[hariz91]

How are you getting those results?

Thanks for the question. I got those results from a simple process I’ve been playing with:

1. I imagine 0 as a source that releases the value 1, one after another.
2. The first 1 appears — it’s just there.
3. When the second 1 appears, the two 1s interact and produce a new value: 2. They form a symmetric sequence: 1, 2, 1.
4. When the third 1 appears, it interacts with the first 2 to produce a new 2, then the old 2 interacts with the old 1 to produce 3, and then interacts with the other 1 to produce another 2. The sequence becomes: 1, 2, 3, 2, 1.
5. Each time a new 1 appears, the same process continues — building a symmetric sequence step by step.

I’m not claiming this is a new discovery — it’s just a number pattern I found interesting and wanted to share.

 

[Gavran]

- 1, [2], 1 → gives 0
- 1, 2, [3], 2, 1 → gives 3
- 1, 2, 3, [4], 3, 2, 1 → gives 8
- 1, 2, 3, 4, [5], 4, 3, 2, 1 → gives 5
...
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1 → gives 63

I’m not sure if this pattern is already known or if it has a name. I’m just curious — has anyone seen something like this before? I’d appreciate any thoughts or references.

0, 3, 8, 15, 24, ... is a known sequence with a general term n²-1 where n=1, 2, 3, ..., but I do not know if it has a particular name.
By the way 1, 2, 3, 4, 5, 4, 3, 2, 1 gives 15.

 

[renormalize]

- 1, 2, 3, 4, [5], 4, 3, 2, 1 → gives 5

Do you mean 15 instead of 5?

 

[pbuk]

1. I imagine 0 as a source that releases the value 1, one after another.
2. The first 1 appears — it’s just there.
3. When the second 1 appears, the two 1s interact and produce a new value: 2. They form a symmetric sequence: 1, 2, 1.
4. When the third 1 appears, it interacts with the first 2 to produce a new 2, then the old 2 interacts with the old 1 to produce 3, and then interacts with the other 1 to produce another 2. The sequence becomes: 1, 2, 3, 2, 1.
5. Each time a new 1 appears, the same process continues — building a symmetric sequence step by step.

This description may be meaningful to you, but when communicating with other people it is not helpful to use phrases like "the two 1s interact and produce a new value: 2".

The way the sequences develop is quite clear from listing them, you don't need any words for this. What is not at all clear though is how you get the numbers on the right (0, 3, 8, 5, ... 63)? Others have noted that these seem to follow the pattern $(n - 1)^2$ where $n$ is the bracketed number (assuming you meant to write 15 instead of 5 - you should take more care), but this doesn't fit your description.

When you have a sequence you are not sure about, a good place to find out more is The On-Line Encyclopedia of Integer Sequences® (OEIS®).