I couldn't find anything about this on the internet and was wondering if there was any information or work done on an idea regarding expanding palindromes? I'm defining expanding palindromes as
A sequence which, when compacted into one number following the original order, continuously creates palindromes as we find the next term in the sequence
An excellent example of what this means would be the ruler function, which would output a sequence like
$$1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5...$$ Which would then compact to the number $1213121412131215...$
We can see that as we progress through each term in the function, we obtain a new palindrome. At first we get to $$1$$ Which is a palindrome seen when $n=1$ (term number is 1) Then the palindrome breaks up and appears again when we get $$1,2,1$$ Which compacts to $121$, a palindrome. Only to break up again when we get
$$1,2,1,3,1,2,1$$ Which compacts to 1213121, which is a new palindrome seen when $n=7$. We can observe that this property lasts forever and occurs whenever $n=2^k-1$ for a positive integer $k$. Although this just applies to the ruler function
Is there any structured idea about this, anything actually formal or is this a new idea? Are there any known sequences that have similar properties? Any ideas would be greatly appreciated!
More Examples with this property (and people who found it):
$f(n) = c$ where $c$ is a constant (John Lou)
You can create as many "expanding palindromes" as you want. The main idea is if you pick $n$ last consecutive terms $a_1, \ldots, a_n$ in the list $x_1, \ldots, x_l, a_1, \ldots, a_n$, you can add $n$ more terms $ a_n, \ldots, a_2,a_1$ to make a palindrome. Keep doing that we get the desired sequence. I particularly don't think there's any interesting property about this type of sequence.