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what's a 6 sided shape - Coaching Toolbox

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 A virologist is studying the combinations of synthetic DNA sequences where two types of nucleotides, A and T, are used. How many 6-nucleotide sequences can be formed where there are exactly two consecutive A's at least once?

📌 To solve this, we first compute the total number of 6-nucleotide sequences using A and T, which is \(2^6 = 64\).

📌 Next, we compute the number of sequences that do **not** contain two consecutive A’s. Let’s denote this count as \(a_n\) for sequences of length \(n\).

📌 We derive a recurrence for \(a_n\):

📌 If the first nucleotide is T, the rest \(n-1\) positions form a valid sequence: \(a_{n-1}\)

📌 If the first is A, the next must be T (to avoid AA), and then \(n-2\) positions follow: \(a_{n-2}\)

📌 So, \(a_n = a_{n-1} + a_{n-2}\), with base cases:

📌 \(a_1 = 2\) (A, T)

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