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.