Closed Form Continued Fraction Expansions of Special Quadratic Irrationals
نویسنده
چکیده
We explore methods for determining the underlying structure of certain classes of continued fractions . The goal is to develop closed form expressions for the continued fractions of many quadratic irrationals. Consider a finite difference equation satisfying: • Gn+1 = anGn + bnGn−1. • an = m, and bn = l for all n, where m, l ∈ N note: m = l = 1 gives the Fibonacci numbers. Let Gn denote the n term of the sequence, and let τ denote the limn→∞ Gn+1 Gn . We derive formulas for Gn Gn−k for certain m, l; in particular, these allow us to determine closed form continued fraction expansions of τ k for any positive integer k.
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