There have been a couple problems I've recently thought about that would benefit from having a certain way of notating the underlying process. Say we have a sequence of sequences $\{a_n\}_m$, $n,m\in\mathbb{N}$, that follow that pattern described below:

$$ \{a_n\}_1=\{a_1^1,a_2^1,a_3^1,a_4^1,\cdots\} $$

where the "exponent" denotes that these elements are part of the first sequence. To continue, the second sequence is almost the same as the first sequence, only some of the terms "divide" (bifurcate), so maybe $a_1^2=a_1^1$ and $a_2^1=a_2^2+a_3^2$ and for $n>3$, $a_n^2 = a_{n-1}^1$

$$ \{a_n\}_2=\{a_1^2,a_2^2,a_3^2,a_4^2,a_5^2,\cdots\}=\{a_1^1,a_2^2,a_3^2,a_3^1,a_4^1,\cdots\} $$

Now, there's also the opposite: some terms might combine (assimilate) together going to the next sequence. So, $a_i^2=a_n^1+a_{n+1}^1$ for some $i, n$. It is sort of like a fractal where the next sequence either draws in or out from the previous sequence. Does anyone have a convenient/compact way of notating this behavior?


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