Palindrome-containing strings

Let F₅(n) be the number of strings s such that:

• s consists only of '0's and '1's,
• s has length at most n, and
• s contains a palindromic substring of length at least 5.

For example, F₅(4) = 0, F₅(5) = 8, F₅(6) = 42 and F₅(11) = 3844.

Let D(L) be the number of integers n such that 5 ≤ n ≤ L and F₅(n) is divisible by 87654321.

For example, D(10⁷) = 0 and D(5·10⁹) = 51.

Find D(10¹⁸).

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter