Using your Head is Permitted

We begin with three definitions. The first two are for familiar mathematical objects, the third is new.

We call a number a palindrome if its digits are the same when read forward-to-backward as when read back-to-front. For example, "131" and "2552" are both palindromes, but "253" is not. Also, "10" is not: you can't add leading zeroes to make a palindrome.

We call a number a square if it equals another number multiplied by itself, so "9" and "16" are both squares, but "35" is not. The number that is to be multiplied by itself to generate the original number is called its root, so "3" is the root of "9" and "4" is the root of "16". (In the present context, "number" refers to the naturals, so "square root of 35" is not a number.)

We call a number a PSP if it satisfies the following 3 conditions:

1. It is a palindrome.
2. It is a square.
3. Its root is a palindrome.

This month's question: as a function of N, how many PSP numbers are there, that are at most N digits in length?

Provide your answer as a closed-form formula. (As usual, with proof of correctness.)

List of solvers: