Seductive Recursive Sequences

Algebra Level 3

\((\alpha_n)\) and \((\beta_n)\) are two sequences defined recursively as \(\alpha_n=2 \alpha_{n-1}+1\) and \(\beta_n=8 \beta_{n-1}+1\) with \(\alpha_0=\beta_0=0.\) Find the ratio

\[ \frac{\alpha_{2016}}{\beta_{672}}.\]

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