I'm working on a problem related to a recursive sequence, and I'm having some trouble. The sequence $S_{n}^{k}(a)$ is defined as follows:
$$S_{n}^{k}(a) = \begin{cases} \sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a \neq 1 \\ \sum_{i=1}^{k} \sqrt{d_i} & \text{if } n > 1, \quad k \in \mathbb{Z}^{+} \end{cases}$$
where $d_i$ represents the $i$-th digit of the previous term $S_{n-1}^{k}(a)$, and $k$ represents the number of digits of $S_{n-1}^{k}(a)$ from which we extract and sum the square roots.
The problem goes as follows:
How many different combinations of $a$, $n$ and $k$ exist such that the following equality is satisfied: $$S_{n}^{k}(a) = S_{n+1}^{k}(b)$$ where $b = a^2$
Can anyone provide guidance or suggestions on how to approach proving this base case? Any insights or hints would be greatly appreciated.
Thank you!
(I am not a native English speaker, so I might be using the incorrect terms)