Cute numbers!

Note that for a cute number to be formed, there needs to be a $2$ at every alternate place, starting from either the first or second digit. So there would be $8$ places to fill either $1$ or $3$ . Meaning, that there are $2^8 \times 2$ possible cute numbers ( $2^8$ starting with $2$ and $2^8$ not starting with $2$ ).

Of each of these, note that for each cute number there would exist another cute number where each $1$ is replaced by $3$ and each $3$ is replaced by $1$ . Adding both of these, you would get $4444444444444444$ ( $16$ -digit $4$ 's). Since there are $2^9$ cute numbers, there would be $\frac{2^9}{2}$ pairs of opposite numbers.

Meaning the sum would be $4444444444444444 \times 2^8$ . We can factor out $4$ from the first term leaving us with $2^{10} \times 1111111111111111$ , which is in the desired form.