We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation G n + 1 = m G n + ℓ G n - 1 with G 0 = 0 , G 1 = 1 , m a positive integer, and ℓ = ± 1 (note that m = ℓ = 1 gives the Fibonacci numbers). Let φ m , ℓ = lim n → ∞ G n / G n - 1 . We find simple closed form continued fraction expansions for φ m , ℓ k for
... [Show full abstract] any integer k by exploiting elementary properties of the recurrence relation and continued fractions.