Brilliant Binary Being Batty

A brilliant binary sequence $$B_k$$ is a sequence of $$k$$ digits made of 0's and 1's such that each digit is adjacent to at least one 1.

Let $$S_k$$ be the number of brilliant binary sequences of length $$k$$. A positive integer $$n$$ is called batty if $$S_n$$ is divisible by 20. Find the smallest $$n > 2$$ that is batty.

×