Find a recurrence relation for the number of binary strings of length “n” that do not contain three of the same number in a row. How do I approach this problem?

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Find a recurrence relation for the number of binary strings of length “n” that do not contain three of the same number in a row. How do I approach this problem?

Mathematics
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I'd start off by creating the sequence that counts the number of strings that DO contain three of the same number in a row. Call this sequence \(b_n\), then clearly \(b_1,b_2=0\), while \(b_3=2\) (since you can have \(111\) or \(000\)). Then whatever this sequence will be, let \(a_n\) denote the number of strings that DO NOT contain the three-in-a-row pattern, so \(a_n=2^n-b_n\). All this to say you'd find a closed form for \(a_n\), then make a recurrence relation out of that.
I've got to close this question and ask an easier one so I understand the foundation of recurrence relationships. I'm totally confused.
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Actually, here's a more direct way of finding what you want. Suppose \(x\) is a binary string of length \(n\). This string can end in \(1\), \(0\), \(11\), or \(00\). (We're basically considering the cases where the last two digits are distinct or the same consecutive digit.) So letting \(a_n\) be the total number of possible \(n\)-length binary strings without \(111\) or \(000\), we'll let \(b_n\) denote the number of such strings that end in \(1\), \(c_n\) the number of such strings ending in \(0\), \(d_n\) for those ending in \(11\), and \(e_n\) for those ending in \(00\). In other words, we have the following relation: \[a_n=b_n+c_n+d_n+e_n\] What we want to do is write \(b_n,c_n,d_n,e_n\) in terms of previous terms in the sequence \(\{a_n\}\). Does that make sense so far?

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