The Lucas numbers, named after François-Eduoard-Anatole Lucas (see Chapter 7 for a biography), are defined recursively by
with L1 = 1 and L2 = 3. They satisfy the same recurrence relation as the Fibonacci numbers, but the two initial values are different.
Show that the nth Lucas number Ln is the sum of the (n + 1)st and (n − 1)st Fibonacci numbers, fn+1 and fn − 1, respectively.
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