Fibonacci Residue Observation

[David Wilson]

Let a(n) be the number of distinct residues of Fibonacci numbers modulo 
Fib(n).  It then appears that

a(1) = 1, a(2) = 1, a(3) = 2,
a(2k) = 3k-3  (k >= 2)
a(2k+1) = 4k-3 (k >= 2)

For example, a(10) = 12, meaning there are 12 residues of Fibonacci numbers 
modulo 55 = Fib(10).
Similarly, there would be only 147 Fibonacci residues modulo 
354224848179261915075 = Fib(100).

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- David Wilson