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📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 $a_n = b_n + c_n + d_n$, but easier: define based on how the string ends.

📌 Use states:

📌 $S_n$: number of valid strings of length $n$ ending in 3

📌 $T_n$: ending in exactly one 2 (i.e., last is 2, previous ≠ 2)

📌 $U_n$: ending in exactly two consecutive 2s (i.e., last two are 22, but not 222)

📌 Recurrences:

📌 $S_n = S_{n-1} + T_{n-1} + U_{n-1}$ (append 3 to any valid string)

📌 $T_n = S_{n-1}$ (append 2 to a string ending in 3)

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