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$$\sqrt{2} = \frac{1}{1} + \frac{2}{5} + \frac{2}{145} + \frac{2}{4,901} + \frac{2}{166,465} + \frac{2}{5,654,885} + \frac{2}{192,099,601} + \frac{2}{6,525,731,525} + \frac{2}{221,682,772,225} + \frac{2}{7,530,688,524,101} + \frac{2}{255,821,727,047,185} + \cdots$$

Let $$d_i$$ be the denominator of the $$i^{th}$$ term of this series. Then $d_{i+1} = 35d_i - 35 d_{i-1} + d_{i-2}$.