\sqrt{2}=1+\frac{1}{\lfloor\sqrt{2}+1\rfloor+\sqrt{2}+1-\lfloor\sqrt{2}+1\rfloor}=1+\frac{1}{2+\sqrt{2}+1-2}=1+\frac{1}{2+\frac{1}{\frac{1}{\sqrt{2}-1}}}=1+\frac{1}{2+\frac{1}{\sqrt{2}+1}}