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yes, LinearFibonacci() is O(n), but you run it m number of times which means the total computation is O(mn) time. And on each run of LinearFibonacci() m = n so that's O(n^2). Why should that be a surprise?
Also, Richie is correct: if you cached your previous results, it would be O(n). You've only come up with an O(n^2) algorithm for a problem that could be solved in O(n) time.
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