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The Lucas numbers satisfy the recurrence relation $L_{n} = L_{n−1} + L_{n−2},$ and the initial conditions $L_{0} = 2$ and $L_{1} = 1.$ 

  1. Show that $L_{n} = f_{n−1} + f_{n+1}\: \text{for}\: n = 2, 3,\dots,$ where fn is the $n^{\text{th}}$ Fibonacci number.
  2. Find an explicit formula for the Lucas numbers.
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