RMO 2025 K6

Define the sequence $(a_0, a_1, a_2, \ldots)$ as follows: $a_0 = 49$ and $a_{n} = 10^{2^n} a_{n-1} - 1$ for $n \ge 1$ . Show that $s(a_n^2) = n^2 + n + 7$ for all $n \ge 0$ , where $s(m)$ denotes the sum of digits in base 10 representation of a nonnegative integer $m$ .