We define a recurrence:

February 22, 2026

We define a recurrence:

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Solution: First, compute the total number of 8-digit numbers using only digits 3 and 4:
2^8 = 256
Now count the number of such numbers that do **not** contain two consecutive 3s. Let $a_n$ be the number of valid sequences of length $n$ with no two consecutive 3s, where each digit is 3 or 4.
If the first digit is 4, the remaining $n-1$ digits form a valid sequence: $a_{n-1}$
If the first digit is 3, the next must be 4, and the rest $n-2$ digits form a valid sequence: $a_{n-2}$
a_1 = 2 \quad (\text{3 or 4}),\quad a_2 = 3 \quad (\text{34, 43, 44})

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